Question
3.4.5 A coin is tossed repeatedly until either two successive heads appear or two successive tails appear. Suppose the first coin toss results in a head. Find the probability that the game ends with two successive tails
Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references.
Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.

This is only a preview of the solution.
Please use the purchase button to see the entire solution.
By purchasing this solution you'll be able to access the following files:
Solution.pdf
Purchase Solution
$5.00
Google Pay
Amazon
Paypal
Mastercard
Visacard
Discover
Amex
View Available Mathematics Tutors 529 tutors matched
ionut
Ionut
(ionut)
Hi! MSc Applied Informatics & Computer Science Engineer. Practical experience in many CS & IT branches.Research work & homework
5/5 (5,654+ sessions)
2 hours avg response
Leo
Leo
(Leo)
Hi! I have been a professor in New York and taught in a math department and in an applied math department.
4.9/5 (5,652+ sessions)
2 hours avg response
Pranay
(math1983)
Ph.D. in mathematics and working as an Assistant Professor in University. I can provide help in mathematics, statistics and allied areas.
4.6/5 (5,512+ sessions)
1 hour avg response

Similar Homework Solutions

8.1.0PHP Version432msRequest Duration45MBMemory UsageGET college-homework-library/{category}/{subject}/{id}Route
    • Booting (279ms)time
    • Application (154ms)time
    • 1 x Booting (64.44%)
      279ms
      1 x Application (35.55%)
      154ms
      • Illuminate\Routing\Events\Routing (1.39ms)
      • Illuminate\Routing\Events\RouteMatched (518μs)
      • Illuminate\Foundation\Events\LocaleUpdated (4.06ms)
      • eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (208μs)
      • eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (203μs)
      • Illuminate\Database\Events\ConnectionEstablished (1.12ms)
      • Illuminate\Database\Events\StatementPrepared (20.34ms)
      • Illuminate\Database\Events\QueryExecuted (995μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (132μs)
      • eloquent.booting: App\Models\Subject (125μs)
      • eloquent.booted: App\Models\Subject (63μs)
      • Illuminate\Database\Events\StatementPrepared (1.6ms)
      • Illuminate\Database\Events\QueryExecuted (1.07ms)
      • eloquent.retrieved: App\Models\Subject (91μs)
      • eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (101μs)
      • eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (40μs)
      • Illuminate\Database\Events\StatementPrepared (593μs)
      • Illuminate\Database\Events\QueryExecuted (704μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (62μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (14μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (8μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (6μs)
      • eloquent.booting: App\Models\SubjectCat (223μs)
      • eloquent.booted: App\Models\SubjectCat (39μs)
      • Illuminate\Database\Events\StatementPrepared (617μs)
      • Illuminate\Database\Events\QueryExecuted (764μs)
      • eloquent.retrieved: App\Models\SubjectCat (67μs)
      • Illuminate\Cache\Events\CacheHit (9.83ms)
      • Illuminate\Cache\Events\CacheMissed (191μs)
      • Illuminate\Database\Events\StatementPrepared (871μs)
      • Illuminate\Database\Events\QueryExecuted (38.09ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (113μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (18μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (9μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (8μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (6μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (6μs)
      • Illuminate\Database\Events\StatementPrepared (717μs)
      • Illuminate\Database\Events\QueryExecuted (3.42ms)
      • eloquent.retrieved: App\Models\Subject (96μs)
      • Illuminate\Cache\Events\KeyWritten (820μs)
      • Illuminate\Database\Events\StatementPrepared (2.6ms)
      • Illuminate\Database\Events\QueryExecuted (2.7ms)
      • Illuminate\Database\Events\StatementPrepared (764μs)
      • Illuminate\Database\Events\QueryExecuted (933μs)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (49μs)
      • Illuminate\Database\Events\StatementPrepared (771μs)
      • Illuminate\Database\Events\QueryExecuted (1.35ms)
      • eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (40μs)
      • Illuminate\Cache\Events\CacheHit (576μs)
      • creating: homework.show (262μs)
      • composing: homework.show (115μs)
      • creating: components.breadcrumbs (1.02ms)
      • composing: components.breadcrumbs (132μs)
      • Illuminate\Database\Events\StatementPrepared (1.06ms)
      • Illuminate\Database\Events\QueryExecuted (763μs)
      • eloquent.retrieved: App\Models\SubjectCat (68μs)
      • Illuminate\Cache\Events\CacheHit (3.01ms)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (133μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (151μs)
      • Illuminate\Cache\Events\CacheHit (139μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (154μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (137μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (349μs)
      • Illuminate\Cache\Events\CacheHit (146μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (281μs)
      • Illuminate\Cache\Events\CacheHit (157μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (121μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (122μs)
      • Illuminate\Cache\Events\CacheHit (235μs)
      • Illuminate\Cache\Events\CacheHit (257μs)
      • Illuminate\Cache\Events\CacheHit (132μs)
      • Illuminate\Cache\Events\CacheHit (125μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (121μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (321μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (127μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (132μs)
      • Illuminate\Cache\Events\CacheHit (100μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (132μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (102μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (121μs)
      • Illuminate\Cache\Events\CacheHit (130μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (299μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (294μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (457μs)
      • Illuminate\Cache\Events\CacheHit (147μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (137μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (285μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (123μs)
      • Illuminate\Cache\Events\CacheHit (98μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (97μs)
      • Illuminate\Cache\Events\CacheHit (270μs)
      • Illuminate\Cache\Events\CacheHit (293μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (267μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (127μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (609μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (98μs)
      • Illuminate\Cache\Events\CacheHit (154μs)
      • Illuminate\Cache\Events\CacheHit (102μs)
      • Illuminate\Cache\Events\CacheHit (229μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (146μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (133μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (97μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (129μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (88μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (87μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (93μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (98μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (130μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (93μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (122μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (98μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (102μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (93μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (588μs)
      • Illuminate\Cache\Events\CacheHit (104μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (98μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (143μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (93μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (88μs)
      • Illuminate\Cache\Events\CacheHit (102μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (103μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (141μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (93μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (90μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (97μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (89μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (89μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (125μs)
      • Illuminate\Cache\Events\CacheHit (92μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (129μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (138μs)
      • creating: site.layouts.app (343μs)
      • composing: site.layouts.app (17μs)
      • creating: components.canonical (390μs)
      • composing: components.canonical (73μs)
      • creating: components.open-graph (135μs)
      • composing: components.open-graph (47μs)
      • creating: site.headers.header (204μs)
      • composing: site.headers.header (46μs)
      • Illuminate\Cache\Events\CacheHit (1.29ms)
      • creating: components.footer (66μs)
      • composing: components.footer (67μs)
      • Illuminate\Cache\Events\CacheHit (598μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (142μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (149μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • creating: components.forms.contact-us (192μs)
      • composing: components.forms.contact-us (91μs)
      • creating: components.forms.get-started (105μs)
      • composing: components.forms.get-started (44μs)
      • creating: components.forms.free-tool-download (84μs)
      • composing: components.forms.free-tool-download (40μs)
      • creating: components.forms.claim-free-worksheet (80μs)
      • composing: components.forms.claim-free-worksheet (38μs)
      • creating: components.forms.tutor-subscription-waitlist (79μs)
      • composing: components.forms.tutor-subscription-waitlist (37μs)
      • creating: components.forms.tutor-subscription-join (77μs)
      • composing: components.forms.tutor-subscription-join (37μs)
      • creating: components.forms.tutor-support (99μs)
      • composing: components.forms.tutor-support (40μs)
      • 321 x Illuminate\Cache\Events\CacheHit (12.77%)
        55.21ms
        10 x Illuminate\Database\Events\QueryExecuted (11.75%)
        50.80ms
        10 x Illuminate\Database\Events\StatementPrepared (6.93%)
        29.95ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.94%)
        4.06ms
        1 x Illuminate\Routing\Events\Routing (0.32%)
        1.39ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.26%)
        1.12ms
        1 x creating: components.breadcrumbs (0.24%)
        1.02ms
        1 x Illuminate\Cache\Events\KeyWritten (0.19%)
        820μs
        1 x Illuminate\Routing\Events\RouteMatched (0.12%)
        518μs
        1 x creating: components.canonical (0.09%)
        390μs
        1 x creating: site.layouts.app (0.08%)
        343μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.07%)
        292μs
        1 x creating: homework.show (0.06%)
        262μs
        1 x eloquent.booting: App\Models\SubjectCat (0.05%)
        223μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        208μs
        1 x creating: site.headers.header (0.05%)
        204μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        203μs
        1 x creating: components.forms.contact-us (0.04%)
        192μs
        1 x Illuminate\Cache\Events\CacheMissed (0.04%)
        191μs
        2 x eloquent.retrieved: App\Models\Subject (0.04%)
        187μs
        6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        179μs
        2 x eloquent.retrieved: App\Models\SubjectCat (0.03%)
        135μs
        1 x creating: components.open-graph (0.03%)
        135μs
        1 x composing: components.breadcrumbs (0.03%)
        132μs
        1 x eloquent.booting: App\Models\Subject (0.03%)
        125μs
        1 x composing: homework.show (0.03%)
        115μs
        1 x creating: components.forms.get-started (0.02%)
        105μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        101μs
        1 x creating: components.forms.tutor-support (0.02%)
        99μs
        1 x composing: components.forms.contact-us (0.02%)
        91μs
        1 x creating: components.forms.free-tool-download (0.02%)
        84μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        80μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        79μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        77μs
        1 x composing: components.canonical (0.02%)
        73μs
        1 x composing: components.footer (0.02%)
        67μs
        1 x creating: components.footer (0.02%)
        66μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        63μs
        1 x composing: components.open-graph (0.01%)
        47μs
        1 x composing: site.headers.header (0.01%)
        46μs
        1 x composing: components.forms.get-started (0.01%)
        44μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        40μs
        1 x composing: components.forms.free-tool-download (0.01%)
        40μs
        1 x composing: components.forms.tutor-support (0.01%)
        40μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        39μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        38μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        37μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        37μs
        1 x composing: site.layouts.app (0%)
        17μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      10 statements were executed, 4 of which were duplicates, 6 unique. Show only duplicated69.62ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '63254' limit 1
        20.6mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 63254
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (60)
        1.15mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (63254)
        890μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        970μstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 63254 and `subject` = 60 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        37.48mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 63254
        • 1: 60
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (60)
        3.56mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 63254 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        2.13mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 63254
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 63254 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        940μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 63254
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 63254 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        910μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 63254
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        990μstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      6HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
      App\Models\SubjectCat
      2SubjectCat.php
          _token
          35bf9SQKG3XXGuiOMVlOL5meyfB7jY2Mj3V3NmAi
          utm_source
          direct
          redirectUrl
          /college-homework-library/Mathematics/Mathematics-Other/63254
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Mathematics/Mat...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Mathematics/Mathematics-Other/63254
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
          array:21 [ "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "x-amzn-trace-id" => array:1 [ 0 => "Root=1-67f3bceb-28c2dde431e967634d0cd79a" ] "host" => array:1 [ 0 => "staging.dev.24houranswers.com" ] "x-forwarded-port" => array:1 [ 0 => "443" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "3.15.209.38" ] "content-length" => array:1 [ 0 => "" ] "content-type" => array:1 [ 0 => "" ] ]
          request_cookies
          []
          
          response_headers
          0 of 0
          array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Mon, 07 Apr 2025 11:54:19 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IkFxWVlNNDB3ZjM4OC9hZVhVdXZKUGc9PSIsInZhbHVlIjoiZE5TQlV3OHNOT0VOMGhwQmVKQnJBWEtpNlVOK2M3STRVTEZJaXhnZDJUN3hEMVFIangxNENGaCtqeVZ5TU5Wc2p3b3JoY1NKeHBuOXdyR2FtRC9ZKzl3bUtVTmRKNW52RHc4cHdGY2VJUlpWV1h0VDdvL1UwRFBPRGYwWHowYzkiLCJtYWMiOiJiMDQyOTEyOTg0ZTk2NTZhNGE5N2NlMmFkOTI5ZTBiNzkwMjVlZDdiZjdiYWEyNTYwOWNjMzc4OGE0MDE0ZGExIiwidGFnIjoiIn0%3D; expires=Mon, 07 Apr 2025 13:54:19 GMT; Max-Age=7200; path=/; domain=.24houranswers.com; samesite=laxXSRF-TOKEN=eyJpdiI6IkFxWVlNNDB3ZjM4OC9hZVhVdXZKUGc9PSIsInZhbHVlIjoiZE5TQlV3OHNOT0VOMGhwQmVKQnJBWEtpNlVOK2M3STRVTEZJaXhnZDJUN3hEMVFIangxNENGaCtqeVZ5TU5Wc2p3b3JoY" 1 => "24houranswers_session=eyJpdiI6InZ3OTNjSG1nSUNiM1FWM2tDNGpHQlE9PSIsInZhbHVlIjoiUmZkMlI5djBOVUpMT1ZrK0tMR2M2UmppZHBIcGN5OGJ3TGd5cmhFL1dwOUpyRHpkdkNoajFVZlBiaU1wRzYwbDRXRkFHczN5UmRUcGl3TkU5MFJNaFhjM1RGaGFIVnZyYkE2VUNvYnRwZDRWY0ZuSWJBVWVJaS9VSWVYWko4bTIiLCJtYWMiOiJmZjA0NTY1YWE0MjMyMjIwMDllOThlYjZkYjE2M2VkMDllMTk0MmJiMTNhMWU0MzRhNWQ3MTJmOTc5OTZkY2NhIiwidGFnIjoiIn0%3D; expires=Mon, 07 Apr 2025 13:54:19 GMT; Max-Age=7200; path=/; domain=.24houranswers.com; httponly; samesite=lax24houranswers_session=eyJpdiI6InZ3OTNjSG1nSUNiM1FWM2tDNGpHQlE9PSIsInZhbHVlIjoiUmZkMlI5djBOVUpMT1ZrK0tMR2M2UmppZHBIcGN5OGJ3TGd5cmhFL1dwOUpyRHpkdkNoajFVZlBiaU1wRz" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IkFxWVlNNDB3ZjM4OC9hZVhVdXZKUGc9PSIsInZhbHVlIjoiZE5TQlV3OHNOT0VOMGhwQmVKQnJBWEtpNlVOK2M3STRVTEZJaXhnZDJUN3hEMVFIangxNENGaCtqeVZ5TU5Wc2p3b3JoY1NKeHBuOXdyR2FtRC9ZKzl3bUtVTmRKNW52RHc4cHdGY2VJUlpWV1h0VDdvL1UwRFBPRGYwWHowYzkiLCJtYWMiOiJiMDQyOTEyOTg0ZTk2NTZhNGE5N2NlMmFkOTI5ZTBiNzkwMjVlZDdiZjdiYWEyNTYwOWNjMzc4OGE0MDE0ZGExIiwidGFnIjoiIn0%3D; expires=Mon, 07-Apr-2025 13:54:19 GMT; domain=.24houranswers.com; path=/XSRF-TOKEN=eyJpdiI6IkFxWVlNNDB3ZjM4OC9hZVhVdXZKUGc9PSIsInZhbHVlIjoiZE5TQlV3OHNOT0VOMGhwQmVKQnJBWEtpNlVOK2M3STRVTEZJaXhnZDJUN3hEMVFIangxNENGaCtqeVZ5TU5Wc2p3b3JoY" 1 => "24houranswers_session=eyJpdiI6InZ3OTNjSG1nSUNiM1FWM2tDNGpHQlE9PSIsInZhbHVlIjoiUmZkMlI5djBOVUpMT1ZrK0tMR2M2UmppZHBIcGN5OGJ3TGd5cmhFL1dwOUpyRHpkdkNoajFVZlBiaU1wRzYwbDRXRkFHczN5UmRUcGl3TkU5MFJNaFhjM1RGaGFIVnZyYkE2VUNvYnRwZDRWY0ZuSWJBVWVJaS9VSWVYWko4bTIiLCJtYWMiOiJmZjA0NTY1YWE0MjMyMjIwMDllOThlYjZkYjE2M2VkMDllMTk0MmJiMTNhMWU0MzRhNWQ3MTJmOTc5OTZkY2NhIiwidGFnIjoiIn0%3D; expires=Mon, 07-Apr-2025 13:54:19 GMT; domain=.24houranswers.com; path=/; httponly24houranswers_session=eyJpdiI6InZ3OTNjSG1nSUNiM1FWM2tDNGpHQlE9PSIsInZhbHVlIjoiUmZkMlI5djBOVUpMT1ZrK0tMR2M2UmppZHBIcGN5OGJ3TGd5cmhFL1dwOUpyRHpkdkNoajFVZlBiaU1wRz" ] ]
          session_attributes
          0 of 0
          array:6 [ "_token" => "35bf9SQKG3XXGuiOMVlOL5meyfB7jY2Mj3V3NmAi" "utm_source" => "direct" "redirectUrl" => "/college-homework-library/Mathematics/Mathematics-Other/63254" "_previous" => array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Mathematics/Mathematics-Other/63254" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]