Transcript Document 7257414

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.






1) Splash Mtn.
7) Roger Rabbit’s Car Toon Spin
2) Mickey’s House
8) Pirates of the Caribbean
3) Big Thunder Mtn. 9) Star Tours
4) Peter Pan’s Flight 10) Haunted Mansion
5) Indiana Jones
11) Matterhorn Mtn.
6) Minnie’s House
12) Space Mtn.
Want to Avoid Long
Lines? Some Attendance
Figures
Source: www.scottware.com.au/theme/feature/atend.htm
 Weekends are the most crowded days at Disneyland
 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.
 September-mid-December (excluding Labor Day,
Columbus Day, and Thanksgiving weekends) are
the least crowded
Problem Time!
 1) Suppose a
family of four
wants to make
four trips to
Disneyland when
it is least
crowded in 2009.
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:
 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.
Solution Cont’d
 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?
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.
 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
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?
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.
 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
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.
 There are 12 popular rides and (for simplicity’s sake) five
lands, so the breakdown is as follows:





Fantasyland: 2/12 attractions
Tomorrowland: 2/12 attractions
Toontown: 3/12 attractions
New Orleans Square/Critter Country: 3/12 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?
Peter Pan
0.2
P (getting sick)
0.5
0.8
Fantasylan
d
P (staying fine)
P (getting sick)
0.7
0.5
Matterhorn
0.3
P (staying fine)
Fantasyland Solution
 Multiply the probabilities of riding the rides and
getting sick. Add them to get your final answer.
Peter Pan
0.2
P (getting sick)
0.5
0.8
Fantasylan
d
P (staying fine)
P (getting sick)
0.7
0.5
Matterhorn
0.3
P (staying fine)
Fantasyland Solution Cont’d
 0.8 * 0.5 + 0.3 * 0.5 =
0.55 chance of
staying fine
Tree Diagram Problem:
New Orleans/Critter
Country
*P(GS) = getting sick. P (F) = staying fine.
 If the family gets sick in New Orleans
Square/Critter Country, what is the probability that
Pirates of the Caribbean got them sick?
P (GS)
0.4
Pirates of the Caribbean
0.6
0.4
New Orleans
Square/Critter
Country
0.4
Splash Mountain
0.2
0.5
Haunted Mansion
0.5
0.9
P (F)
P (GS)
0.1
P (GS)
P (F)
P (F)
New Orleans/Critter
Country Solution
*P(GS) = getting sick. P (F) = staying fine.
 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. 0.4
P (GS)
Pirates of the Caribbean
0.6
0.4
New Orleans
Square/Critter
Country
0.4
Splash Mountain
0.2
0.5
Haunted Mansion
0.5
0.9
P (F)
P (GS)
0.1
P (GS)
P (F)
P (F)
New Orleans/Critter
Country Solution Cont’d
 (0.4*0.4)/(0.9*0.4 +
0.4*0.4 + 0.5*0.2) =
0.2581 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?
P (GS)
0.1
Mickey’s House
0.4
Toontown
0.3
0.9
0.2
Minnie’s House
0.3
P (GS)
0.8
P (GS)
0.8
Roger Rabbit
P (F)
0.2
P (F)
P (F)
Toontown Solution
 Just read it off the tree diagram. Pick the attraction
with the lowest sickness rate.
P (GS)
0.1
Mickey’s House
0.4
Toontown
0.3
0.9
0.2
Minnie’s House
0.3
P (GS)
0.8
P (GS)
0.8
Roger Rabbit
P (F)
0.2
P (F)
P (F)
Toontown Solution Cont’d
 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?
0.8
Space Mountain
P (GS)
0.2
0.7
P (F)
Tomorrowland
0.3
P (GS)
0.7
Star Tours
0.3
P (F)
Tomorrowland Solution
 Read it off the tree diagram, and pick the ride with the higher
probability of being visited.
0.8
Space Mountain
P (GS)
0.2
0.7
P (F)
Tomorrowland
0.3
P (GS)
0.7
Star Tours
0.3
P (F)
Tomorrowland Solution Cont’d
 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?
P (GS)
0.6
Indiana Jones
0.5
Adventure/Fro
ntierland
0.4
P (F)
0.5
P (GS)
0.5
Thunder
Mountain
0.5
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.
P (GS)
0.6
Indiana Jones
0.5
Adventure/Fro
ntierland
0.4
P (F)
0.5
P (GS)
0.5
Thunder
Mountain
0.5
P (F)
Adventure/Frontierland
Solution
 0.5*0.5 + 0.6*0.5 =
.55 probability of
getting sick on
either ride.