Disneyland and Probability Zachary Kuiland MAT 119 Honors Project

Most Popular Rides According to one poll by About.com, the most popular rides at Disneyland are as follows. According to one poll by About.com, the most popular rides at Disneyland are as follows. 1) Splash Mtn. 7) Roger Rabbit’s Car Toon Spin 1) Splash Mtn. 7) Roger Rabbit’s Car Toon Spin 2) Mickey’s House 8) Pirates of the Caribbean 2) Mickey’s House 8) Pirates of the Caribbean 3) Big Thunder Mtn. 9) Star Tours 3) Big Thunder Mtn. 9) Star Tours 4) Peter Pan’s Flight 10) Haunted Mansion 4) Peter Pan’s Flight 10) Haunted Mansion 5) Indiana Jones 11) Matterhorn Mtn. 5) Indiana Jones 11) Matterhorn Mtn. 6) Minnie’s House 12) Space Mtn. 6) Minnie’s House 12) Space Mtn.

Want to Avoid Long Lines? Some Attendance Figures Weekends are the most crowded days at Disneyland Weekends are the most crowded days at Disneyland Tuesdays, Wednesdays, and Thursdays are the least crowded days at the park. Tuesdays, Wednesdays, and Thursdays are the least crowded days at the park. The holidays are the most crowded time of year overall (60-90 minute wait time for major rides), followed by spring break and three day weekends. The holidays are the most crowded time of year overall (60-90 minute wait time for major rides), followed by spring break and three day weekends. September-mid-December (excluding Labor Day, Columbus Day, and Thanksgiving weekends) are the least crowded September-mid-December (excluding Labor Day, Columbus Day, and Thanksgiving weekends) are the least crowded Source:

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

Solution September through mid-December are the least crowded months, excluding Labor Day and its preceding week, Columbus Day, and Thanksgiving week. If we exclude weekends, Mondays, and Fridays: September through mid-December are the least crowded months, excluding Labor Day and its preceding week, Columbus Day, and Thanksgiving week. If we exclude weekends, Mondays, and Fridays: 11 days in September, 13 in October, 9 in November, and 9 in December, totaling 42 possible opportunities to visit. 11 days in September, 13 in October, 9 in November, and 9 in December, totaling 42 possible opportunities to visit. The family wants to go four times, and the days they go do not matter. Thus, we would want to use the choose function. The family wants to go four times, and the days they go do not matter. Thus, we would want to use the choose function.

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

Problem #2 What is the probability that the family will go exactly once every month? What is the probability that the family will go exactly once every month?

Solution #2 You CHOOSE one day from each month to go, and you divide the remaining result by all the possible ways you could go. You CHOOSE one day from each month to go, and you divide the remaining result by all the possible ways you could go. Since you’re only picking one day from each month to go, you can just write the numbers of the days of the month instead of the choose function. [11 instead of C(11, 1), for example] Since you’re only picking one day from each month to go, you can just write the numbers of the days of the month instead of the choose function. [11 instead of C(11, 1), for example]

Solution #2 Cont’d 11*13*9*9/111,930 =.1034 probability that they will go exactly once each month 11*13*9*9/111,930 =.1034 probability that they will go exactly once each month

Problem #3 The family wants to visit at least half of the most popular attractions at Disneyland. How many ways are there to do this? The family wants to visit at least half of the most popular attractions at Disneyland. How many ways are there to do this?

Solution #3 They want to ride at least SIX rides, so you would use the choose function for numbers 6-12 and then add your results together. They want to ride at least SIX rides, so you would use the choose function for numbers 6-12 and then add your results together. Why? They can visit six attractions in C(12, 6) ways, then seven attractions in C(12, 7) ways, and so on. Why? They can visit six attractions in C(12, 6) ways, then seven attractions in C(12, 7) ways, and so on.

Solution #3 Cont’d C(12, 6) + C(12, 7) + C(12, 8) + C(12, 9) + C(12, 10) + C(12, 11) +C(12, 12) = 2,510 ways to ride at least half the popular rides at Disneyland C(12, 6) + C(12, 7) + C(12, 8) + C(12, 9) + C(12, 10) + C(12, 11) +C(12, 12) = 2,510 ways to ride at least half the popular rides at Disneyland

Tree Diagram Problems The next set of problems applies a layer of tree diagrams to the most popular rides and lands at Disneyland. The first layer is the attraction that they’re at, and the second is the probability of the family getting sick on the rides. The next set of problems applies a layer of tree diagrams to the most popular rides and lands at Disneyland. The first layer is the attraction that they’re at, and the second is the probability of the family getting sick on the rides. There are 12 popular rides and (for simplicity’s sake) five lands, so the breakdown is as follows: There are 12 popular rides and (for simplicity’s sake) five lands, so the breakdown is as follows: Fantasyland: 2/12 attractions Fantasyland: 2/12 attractions Tomorrowland: 2/12 attractions Tomorrowland: 2/12 attractions Toontown: 3/12 attractions Toontown: 3/12 attractions New Orleans Square/Critter Country: 3/12 attractions New Orleans Square/Critter Country: 3/12 attractions Adventureland/Frontierland: 2/12 attractions Adventureland/Frontierland: 2/12 attractions

Tree Diagram Problems — Fantasyland The family’s first stop in Disneyland is Fantasyland. If they ride both of Fantasyland’s most popular rides, what is the probability that they will stay fine? The family’s first stop in Disneyland is Fantasyland. If they ride both of Fantasyland’s most popular rides, what is the probability that they will stay fine? Fantasylan d Peter Pan Matterhorn 0.5 P (getting sick) P (getting sick) P (staying fine)

Fantasyland Solution Multiply the probabilities of riding the rides and getting sick. Add them to get your final answer. Multiply the probabilities of riding the rides and getting sick. Add them to get your final answer. Fantasylan d Peter Pan Matterhorn 0.5 P (getting sick) P (getting sick) P (staying fine)

Fantasyland Solution Cont’d 0.8 * * 0.5 = 0.55 chance of staying fine 0.8 * * 0.5 = 0.55 chance of staying fine

Tree Diagram Problem: New Orleans/Critter Country If the family gets sick in New Orleans Square/Critter Country, what is the probability that Pirates of the Caribbean got them sick? If the family gets sick in New Orleans Square/Critter Country, what is the probability that Pirates of the Caribbean got them sick? New Orleans Square/Critter Country Haunted Mansion Pirates of the Caribbean Splash Mountain P (GS) P (F) *P(GS) = getting sick. P (F) = staying fine.

New Orleans/Critter Country Solution Multiply together the probability of riding Pirates with the probability of getting sick. Divide your answer by the sum of all the probabilities of getting sick on the rides to get your final answer. Multiply together the probability of riding Pirates with the probability of getting sick. Divide your answer by the sum of all the probabilities of getting sick on the rides to get your final answer. New Orleans Square/Critter Country Haunted Mansion Pirates of the Caribbean Splash Mountain P (GS) P (F) *P(GS) = getting sick. P (F) = staying fine.

New Orleans/Critter Country Solution Cont’d (0.4*0.4)/(0.9* * *0.2) = chance of Pirates getting them sick. (0.4*0.4)/(0.9* * *0.2) = chance of Pirates getting them sick.

Tree Diagram Problem: Toontown The family decides to go Toontown to let their stomachs settle a little bit. What attraction should they visit that is least likely to get them sick there? The family decides to go Toontown to let their stomachs settle a little bit. What attraction should they visit that is least likely to get them sick there? P (F) P (GS) Toontown Mickey’s House Minnie’s House Roger Rabbit

Toontown Solution Just read it off the tree diagram. Pick the attraction with the lowest sickness rate. Just read it off the tree diagram. Pick the attraction with the lowest sickness rate. P (F) P (GS) Toontown Mickey’s House Minnie’s House Roger Rabbit

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

Tree Diagram Problem: Tomorrowland After an hour or so to rest up in Toontown, the family heads to Tomorrowland for a chance to go on another thrilling ride. Which ride are they more likely to go on? After an hour or so to rest up in Toontown, the family heads to Tomorrowland for a chance to go on another thrilling ride. Which ride are they more likely to go on? Tomorrowland Star Tours Space Mountain P (F) P (GS) P (F) P (GS)

Tomorrowland Solution Read it off the tree diagram, and pick the ride with the higher probability of being visited. Read it off the tree diagram, and pick the ride with the higher probability of being visited. Tomorrowland Star Tours Space Mountain P (F) P (GS) P (F) P (GS)

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

Tree Diagram Problem: Adventure/Frontierland It’s almost time to head home, and the family has time for both rides in Adventure/Frontierland. What is the probability a ride there gets them sick? It’s almost time to head home, and the family has time for both rides in Adventure/Frontierland. What is the probability a ride there gets them sick? Adventure/Fro ntierland Indiana Jones Thunder Mountain P (GS) P (F)

Adventure/Frontierland Solution Multiply the probabilities of riding the rides with the probability that the family will get sick on them. Add the products together to get your final answer. Multiply the probabilities of riding the rides with the probability that the family will get sick on them. Add the products together to get your final answer Adventure/Fro ntierland Indiana Jones Thunder Mountain P (GS) P (F)

Adventure/Frontierland Solution 0.5* *0.5 =.55 probability of getting sick on either ride. 0.5* *0.5 =.55 probability of getting sick on either ride.