1. Describe how each of these three questions refers to a binomial...

Question
1. Describe how each of these three questions refers to a binomial probability distribution. Determine and explain your answer to each of the three questions. Include the Excel output in your submission.
a. You have invited 10 friends to a cookout. On any given day, the probability that any one of them will elect to eat a hotdog as their entrée is 35%. You have 4 hotdogs. What is the likelihood that you will run out of hotdogs?
b. The weather forecast says that there is a 20% chance of rain each of the 4 hours during which your picnic is scheduled. What is the probability that there will be no rain in 3 of the 4 hours?
c. As you play softball, you know that each of your guests reaches base 40% of the time. What is the likelihood that in their first time at bat, exactly 6 of 10 of your guests will reach base?

2. Select one of menu preference, weather forecasts, or batting average and describe how the binomial probability distribution could be used to aid a commercial business.
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Problem 1 a)

You have invited 10 friends to a cookout. On any given day, the probability that any one of them will elect to eat a hotdog as their entrée is 35%. You have 4 hotdogs. What is the likelihood that you will run out of hotdogs?

Solution:
This is a problem about the binomial distribution.
N= number of trials =10 is the number of participants.
X is the number of successes = number of hotdogs eaten,
and the probability that anyone of the participants eats a hot dog is equal to p = 0.35. We assume that whether anyone eats a hotdog is independent of whether anyone else chooses to eat a hotdog, and that anyone eats at most one hotdog.

We run out of hotdogs when X>4 and so we need to determine the probability P(X>4).
Since X has the binomial distribution: P(X=k)= 10Ck pk(1-p)(10-k) _, with p=0.35.

In Excel we can compute the probability that P(X<=4) as: binom.dist(4,10,0.35,TRUE) and we obtain P(X<=4)= 0.751496. Therefore P(X>4)=1-P(X<=4)= 0.248504, which is then the probability that we run out of hotdogs.
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      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (236μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (207μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheMissed (276μs)
      • Illuminate\Database\Events\StatementPrepared (971μs)
      • Illuminate\Database\Events\QueryExecuted (839μs)
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      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (223μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (264μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (384μs)
      • Illuminate\Cache\Events\CacheHit (456μs)
      • creating: site.layouts.app (583μs)
      • composing: site.layouts.app (24μs)
      • creating: components.canonical (344μs)
      • composing: components.canonical (88μs)
      • creating: components.open-graph (162μs)
      • composing: components.open-graph (71μs)
      • creating: site.headers.header (221μs)
      • composing: site.headers.header (69μs)
      • Illuminate\Cache\Events\CacheHit (1.66ms)
      • creating: components.footer (76μs)
      • composing: components.footer (121μs)
      • Illuminate\Cache\Events\CacheHit (973μs)
      • Illuminate\Cache\Events\CacheMissed (363μs)
      • Illuminate\Database\Events\StatementPrepared (730μs)
      • Illuminate\Database\Events\QueryExecuted (1.07ms)
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      • Illuminate\Cache\Events\KeyWritten (341μs)
      • Illuminate\Cache\Events\CacheHit (235μs)
      • Illuminate\Cache\Events\CacheHit (246μs)
      • Illuminate\Cache\Events\CacheHit (260μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (123μs)
      • Illuminate\Cache\Events\CacheHit (325μs)
      • Illuminate\Cache\Events\CacheHit (329μs)
      • Illuminate\Cache\Events\CacheHit (302μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (289μs)
      • Illuminate\Cache\Events\CacheHit (337μs)
      • Illuminate\Cache\Events\CacheHit (333μs)
      • creating: components.forms.contact-us (251μs)
      • composing: components.forms.contact-us (145μs)
      • creating: components.forms.get-started (190μs)
      • composing: components.forms.get-started (79μs)
      • creating: components.forms.free-tool-download (138μs)
      • composing: components.forms.free-tool-download (152μs)
      • creating: components.forms.claim-free-worksheet (150μs)
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      • creating: components.forms.tutor-subscription-waitlist (168μs)
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      • creating: components.forms.tutor-subscription-join (135μs)
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      • creating: components.forms.tutor-support (127μs)
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      • 312 x Illuminate\Cache\Events\CacheHit (17.84%)
        89.08ms
        19 x Illuminate\Database\Events\QueryExecuted (7.72%)
        38.55ms
        19 x Illuminate\Database\Events\StatementPrepared (6.94%)
        34.66ms
        10 x Illuminate\Cache\Events\CacheMissed (1.66%)
        8.31ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.54%)
        7.71ms
        10 x Illuminate\Cache\Events\KeyWritten (1.33%)
        6.64ms
        1 x Illuminate\Routing\Events\Routing (0.27%)
        1.36ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.22%)
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        583μs
        1 x Illuminate\Routing\Events\RouteMatched (0.11%)
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        1 x eloquent.booting: App\Models\SubjectCat (0.08%)
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        2 x eloquent.retrieved: App\Models\Subject (0.07%)
        359μs
        1 x creating: homework.show (0.07%)
        347μs
        1 x creating: components.canonical (0.07%)
        344μs
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        341μs
        1 x creating: components.forms.contact-us (0.05%)
        251μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
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        8 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
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        1 x creating: site.headers.header (0.04%)
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        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
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        209μs
        1 x creating: components.forms.get-started (0.04%)
        190μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        168μs
        1 x composing: components.breadcrumbs (0.03%)
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        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        163μs
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        162μs
        1 x composing: homework.show (0.03%)
        152μs
        1 x composing: components.forms.free-tool-download (0.03%)
        152μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        150μs
        1 x composing: components.forms.contact-us (0.03%)
        145μs
        1 x creating: components.forms.free-tool-download (0.03%)
        138μs
        1 x composing: components.forms.claim-free-worksheet (0.03%)
        138μs
        1 x creating: components.forms.tutor-subscription-join (0.03%)
        135μs
        1 x creating: components.forms.tutor-support (0.03%)
        127μs
        1 x composing: components.footer (0.02%)
        121μs
        1 x composing: components.canonical (0.02%)
        88μs
        1 x composing: components.forms.get-started (0.02%)
        79μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.02%)
        77μs
        1 x creating: components.footer (0.02%)
        76μs
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        76μs
        1 x composing: components.open-graph (0.01%)
        71μs
        1 x composing: site.headers.header (0.01%)
        69μs
        1 x composing: components.forms.tutor-support (0.01%)
        67μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        62μs
        1 x eloquent.booting: App\Models\Subject (0.01%)
        59μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        59μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        45μs
        1 x composing: site.layouts.app (0%)
        24μs
      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
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      • 1x components.forms.get-startedget-started.blade.phpblade
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      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
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      namespace
      where
      as
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        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
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        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 (8)
        1mstwenty4_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
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        2.86mstwenty4_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
        1.63mstwenty4_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
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        16.93mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
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        • 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 (8)
        1.12mstwenty4_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` = 29651 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        960μstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        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
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        970μstwenty4_siteHomeworkLibrary.php#260
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        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
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        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
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        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
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      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.16mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        890μstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        870μstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.07mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        2.39mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        870μstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        880μstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.21mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.08mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
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      App\Models\HomeworkLibrary\HomeworkLibraryFile
      8HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
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      App\Models\Subject
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