Disneyland and Disneyland and ProbabilityProbabilityZachary KuilandZachary KuilandMAT 119 Honors ProjectMAT 119 Honors Project

Most Popular RidesMost Popular Rides

According to one poll by About.com, the most According to one poll by About.com, the most popular rides at Disneyland are as follows.popular rides at Disneyland are as follows. 1) Splash Mtn.1) Splash Mtn. 7) Roger Rabbit’s Car 7) Roger Rabbit’s Car

Toon SpinToon Spin 2) Mickey’s House 8) Pirates of the 2) Mickey’s House 8) Pirates of the

CaribbeanCaribbean 3) Big Thunder Mtn. 9) Star Tours3) Big Thunder Mtn. 9) Star Tours 4) Peter Pan’s Flight 10) Haunted Mansion4) Peter Pan’s Flight 10) Haunted Mansion 5) Indiana Jones 5) Indiana Jones 11) Matterhorn Mtn. 11) Matterhorn Mtn. 6) Minnie’s House 12) Space Mtn.6) Minnie’s House 12) Space Mtn.

Want to Avoid Long Want to Avoid Long Lines? Some Lines? Some

Attendance FiguresAttendance FiguresWeekends are the most crowded days at Weekends are the most crowded days at

DisneylandDisneyland

Tuesdays, Wednesdays, and Thursdays are the Tuesdays, Wednesdays, and Thursdays are the least crowded days at the park.least crowded days at the park.

The holidays are the most crowded time of year The holidays are the most crowded time of year overall (60-90 minute wait time for major rides), overall (60-90 minute wait time for major rides), followed by spring break and three day weekends.followed by spring break and three day weekends.

September-mid-December (excluding Labor Day, September-mid-December (excluding Labor Day, Columbus Day, and Thanksgiving weekends) are Columbus Day, and Thanksgiving weekends) are the least crowdedthe least crowded

Source: www.scottware.com.au/theme/feature/atend.htm

Problem Time!Problem Time!

1) Suppose a 1) Suppose a family of four family of four wants to make wants to make four trips to four trips to Disneyland Disneyland when it is least when it is least crowded in crowded in 2009. How 2009. How many possible many possible ways are there ways are there to do this?to do this?

SolutionSolution

September through mid-December are the September through mid-December are the least crowded months, excluding Labor Day least crowded months, excluding Labor Day and its preceding week, Columbus Day, and its preceding week, Columbus Day, and Thanksgiving week. If we exclude and Thanksgiving week. If we exclude weekends, Mondays, and Fridays:weekends, Mondays, and Fridays: 11 days in September, 13 in October, 9 in 11 days in September, 13 in October, 9 in

November, and 9 in December, totaling 42 November, and 9 in December, totaling 42 possible opportunities to visit.possible opportunities to visit.

The family wants to go four times, and the The family wants to go four times, and the days they go do not matter. Thus, we days they go do not matter. Thus, we would want to use the choose function.would want to use the choose function.

Solution Cont’dSolution Cont’d

C (42, 4) = 42!/(38!4!) = C (42, 4) = 42!/(38!4!) = 111,930 111,930 ways to make four trips to ways to make four trips to Disneyland.Disneyland.

Problem #2Problem #2

What is the What is the probability probability that the family that the family will go exactly will go exactly once every once every month?month?

Solution #2Solution #2

You CHOOSE one day from each month to go, You CHOOSE one day from each month to go, and you divide the remaining result by all the and you divide the remaining result by all the possible ways you could go.possible ways you could go.

Since you’re only picking one day from each Since you’re only picking one day from each month to go, you can just write the numbers month to go, you can just write the numbers of the days of the month instead of the of the days of the month instead of the choose function. [11 instead of C(11, 1), for choose function. [11 instead of C(11, 1), for example]example]

Solution #2 Cont’dSolution #2 Cont’d

11*13*9*9/111,930 = .1034 probability that 11*13*9*9/111,930 = .1034 probability that they will go exactly once each monththey will go exactly once each month

Problem #3Problem #3

The family The family wants to visit at wants to visit at least half of the least half of the most popular most popular attractions at attractions at Disneyland. How Disneyland. How many ways are many ways are there to do this?there to do this?

Solution #3Solution #3

They want to ride at least SIX rides, so you They want to ride at least SIX rides, so you would use the choose function for numbers 6-would use the choose function for numbers 6-12 and then add your results together. 12 and then add your results together.

Why? They can visit six attractions in C(12, 6) Why? They can visit six attractions in C(12, 6) ways, then seven attractions in C(12, 7) ways, ways, then seven attractions in C(12, 7) ways, and so on.and so on.

Solution #3 Cont’dSolution #3 Cont’d

C(12, 6) + C(12, 7) + C(12, 8) + C(12, 9) + C(12, 6) + C(12, 7) + C(12, 8) + C(12, 9) +

C(12, 10) + C(12, 11) +C(12, 12) C(12, 10) + C(12, 11) +C(12, 12) = 2,510 = 2,510 ways to ride at least half the popular ways to ride at least half the popular rides at Disneylandrides at Disneyland

Tree Diagram ProblemsTree Diagram Problems

The next set of problems applies a layer of tree The next set of problems applies a layer of tree diagrams to the most popular rides and lands at diagrams to the most popular rides and lands at Disneyland. The first layer is the attraction that Disneyland. The first layer is the attraction that they’re at, and the second is the probability of the they’re at, and the second is the probability of the family getting sick on the rides.family getting sick on the rides.

There are 12 popular rides and (for simplicity’s There are 12 popular rides and (for simplicity’s sake) five lands, so the breakdown is as follows:sake) five lands, so the breakdown is as follows: Fantasyland: 2/12 attractionsFantasyland: 2/12 attractions Tomorrowland: 2/12 attractionsTomorrowland: 2/12 attractions Toontown: 3/12 attractionsToontown: 3/12 attractions New Orleans Square/Critter Country: 3/12 attractionsNew Orleans Square/Critter Country: 3/12 attractions Adventureland/Frontierland: 2/12 attractionsAdventureland/Frontierland: 2/12 attractions

Tree Diagram Problems Tree Diagram Problems —Fantasyland —Fantasyland

The family’s first stop in Disneyland is The family’s first stop in Disneyland is Fantasyland. If they ride both of Fantasyland. If they ride both of Fantasyland’s most popular rides, what is the Fantasyland’s most popular rides, what is the probability that they will stay fine?probability that they will stay fine?

Fantasyland

Peter Pan

Matterhorn

0.5

0.5

P (getting sick)

0.3

0.7

0.8

0.2

P (getting sick)

P (staying fine)

P (staying fine)

Fantasyland SolutionFantasyland Solution

Multiply the probabilities of riding the rides Multiply the probabilities of riding the rides and getting sick. Add them to get your final and getting sick. Add them to get your final answer. answer.

Fantasyland

Peter Pan

Matterhorn

0.5

0.5

P (getting sick)

0.3

0.7

0.8

0.2

P (getting sick)

P (staying fine)

P (staying fine)

Fantasyland Solution Fantasyland Solution Cont’dCont’d

0.8 * 0.5 + 0.3 * 0.8 * 0.5 + 0.3 * 0.5 = 0.55 0.5 = 0.55 chance of chance of staying finestaying fine

Tree Diagram Problem: Tree Diagram Problem: New Orleans/Critter New Orleans/Critter

CountryCountry If the family gets sick in New Orleans If the family gets sick in New Orleans

Square/Critter Country, what is the probability Square/Critter Country, what is the probability that Pirates of the Caribbean got them sick?that Pirates of the Caribbean got them sick?

New Orleans Square/Critter Country

Haunted Mansion

Pirates of the Caribbean

Splash Mountain

0.4

0.2

0.6

0.4

0.4

0.5

0.5

0.9

0.1

P (GS)

P (GS)

P (GS)

P (F)

P (F)

P (F)

*P(GS) = getting sick. P (F) = staying fine.

New Orleans/Critter New Orleans/Critter Country SolutionCountry Solution

Multiply together the probability of riding Multiply together the probability of riding Pirates with the probability of getting sick. Pirates with the probability of getting sick. Divide your answer by the sum of all the Divide your answer by the sum of all the probabilities of getting sick on the rides to get probabilities of getting sick on the rides to get your final answer.your final answer.

New Orleans Square/Critter Country

Haunted Mansion

Pirates of the Caribbean

Splash Mountain

0.4

0.2

0.6

0.4

0.4

0.5

0.5

0.9

0.1

P (GS)

P (GS)

P (GS)

P (F)

P (F)

P (F)

*P(GS) = getting sick. P (F) = staying fine.

New Orleans/Critter New Orleans/Critter Country Solution Cont’dCountry Solution Cont’d

(0.4*0.4)/(0.4*0.4)/(0.9*0.4 + (0.9*0.4 + 0.4*0.4 + 0.4*0.4 + 0.5*0.2) = 0.5*0.2) = 0.2581 chance of 0.2581 chance of Pirates getting Pirates getting them sick.them sick.

Tree Diagram Problem: Tree Diagram Problem: ToontownToontown

The family decides to go Toontown to let their The family decides to go Toontown to let their stomachs settle a little bit. What attraction stomachs settle a little bit. What attraction should they visit that is least likely to get should they visit that is least likely to get them sick there?them sick there?

P (F)

P (F)

P (F)

P (GS)

P (GS)

P (GS)

Toontown

Mickey’s House

Minnie’s House

Roger Rabbit 0.2

0.20.3

0.4

0.3

0.8

0.9

0.8

0.1

Toontown SolutionToontown Solution

Just read it off the tree diagram. Pick the Just read it off the tree diagram. Pick the attraction with the lowest sickness rate.attraction with the lowest sickness rate.

P (F)

P (F)

P (F)

P (GS)

P (GS)

P (GS)

Toontown

Mickey’s House

Minnie’s House

Roger Rabbit 0.2

0.20.3

0.4

0.3

0.8

0.9

0.8

0.1

Toontown Solution Toontown Solution Cont’dCont’d

It’s Mickey’s It’s Mickey’s House, with a 0.1 House, with a 0.1 chance of getting chance of getting sick on it.sick on it.

Tree Diagram Problem: Tree Diagram Problem: TomorrowlandTomorrowland

After an hour or so to rest up in Toontown, the family After an hour or so to rest up in Toontown, the family heads to Tomorrowland for a chance to go on another heads to Tomorrowland for a chance to go on another thrilling ride. Which ride are they more likely to go on?thrilling ride. Which ride are they more likely to go on?

Tomorrowland

Star Tours

Space Mountain

P (F)

P (GS)

P (F)

P (GS)0.3

0.3

0.7

0.2

0.8

0.7

Tomorrowland SolutionTomorrowland Solution

Read it off the tree diagram, and pick the ride with the Read it off the tree diagram, and pick the ride with the higher probability of being visited.higher probability of being visited.

Tomorrowland

Star Tours

Space Mountain

P (F)

P (GS)

P (F)

P (GS)0.3

0.3

0.7

0.2

0.8

0.7

Tomorrowland Solution Tomorrowland Solution Cont’dCont’d

They are more They are more likely to visit likely to visit Space Mountain Space Mountain because it has a because it has a 70% chance of 70% chance of being visited being visited over Star Tours.over Star Tours.

Tree Diagram Problem: Tree Diagram Problem: Adventure/FrontierlandAdventure/Frontierland

It’s almost time to head home, and the family It’s almost time to head home, and the family has time for both rides in has time for both rides in Adventure/Frontierland. What is the Adventure/Frontierland. What is the probability a ride there gets them sick?probability a ride there gets them sick?

0.5

0.6

0.4

0.5

0.50.5

Adventure/Frontierland

Indiana Jones

Thunder Mountain

P (GS)

P (GS)

P (F)

P (F)

Adventure/Frontierland Adventure/Frontierland SolutionSolution

Multiply the probabilities of riding the rides Multiply the probabilities of riding the rides with the probability that the family will get with the probability that the family will get sick on them. Add the products together to sick on them. Add the products together to get your final answer.get your final answer.

0.5

0.6

0.4

0.5

0.50.5

Adventure/Frontierland

Indiana Jones

Thunder Mountain

P (GS)

P (GS)

P (F)

P (F)

Adventure/Frontierland Adventure/Frontierland SolutionSolution

0.5*0.5 + 0.5*0.5 + 0.6*0.5 = .55 0.6*0.5 = .55 probability of probability of getting sick on getting sick on either ride.either ride.