Introducing Students to Classic Problems in Probability

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Introducing Students to Classic Problems in
Probability

• Allan Rossman
• Department of Statistics
• Cal Poly San Luis Obispo
• arossman_at_calpoly.edu

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Whats my point?

• Classic problems in probability
• Provide great opportunities for exploring,
  understanding concepts of randomness
• Lend themselves to interactive investigations
• Can be presented at various mathematical levels
• Illustrate mathematical as well as statistical
  ideas
• Create pedagogical alternatives to rolling dice
  and flipping coins for their own sake
• Are intellectually stimulating
• Can be fun!

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What are my goals for this workshop?

• Acquaint you with some classic problems in
  probability
• Demonstrate solutions to these problems
• Using simulation
• Using enumeration
• Provide ideas for activities that you can use
  with your students
• In variety of courses, levels

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What concepts will be introduced?

• Randomness
• Probability
• Simulation
• Relative frequency
• Permutations
• Expected value
• Decision making under uncertainty

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What are these classic problems?

• Problem of the points
• Matching problem
• Collector problem
• Secretary problem

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Problem of the points

• Helped to motive the study of probability
• Discussed in correspondence between Pascal and
  Fermat

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Problem of the points

• A simplified version
• Suppose that Heather and Tom each put in 5 to
  play a game
• They flip a fair coin repeatedly
• If 4 Heads occur before 4 Tails, Heather wins
• If 4 Tails occur before 4 Heads, Tom wins

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Problem of the points

• Outcome H H T H T (3 Heads, 2 Tails)
• Then game is interrupted, cannot be completed!
• How should the prize (10) be divided?
• What might you propose?

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Problem of the points

• My students answers
• Game was not finished, so each takes 5 back
• Heather won 3/5 of the flips, so she gets 6, Tom
  gets 4
• More mathematical approach
• Divide prize proportionally to each players
  probability of winning if the game were continued

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Problem of the points

• How to calculate these probabilities?
• Simulation
• Enumeration
• Ill ask everyone to use a coin to simulate 5
  repetitions of this random process
• Remember that Heather wins with one more Head,
  but Tom needs two Tails to win
• Estimate Heathers probability of winning by
  proportion of times that she wins

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Problem of the points

• Enumeration analysis
• H Heather wins
• TH Heather wins
• TT Tom wins
• Are these equally likely, so Heather has 2/3
  probability of winning?
• No H (prob .5), TH (prob .25), TT (prob .25)
• Heathers probability of winning ¾ .75
• So, Heather should take 7.50, Tom 2.50

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Problem of the points

• Variation What if they were playing to get 10
  heads or tails and were interrupted after 9 heads
  and 8 tails?
• Heather needs 1 more Head, Tom needs 2 more Tails
• Same analysis, same answer!

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Problem of the points

• New variation What if they were playing to 5 and
  had stopped after H H T H T?
• More complicated enumeration Heather wins with
• H H (prob .25)
• H T H (.125)
• H T T H (.0625)
• T H H (.125)
• T H T H (.0625)
• T T H H (.0625)
• Heather has .6875 probability of winning
• Heather should take 6.875

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Matching problem

• Four mothers give birth to baby boys on the same
  night at the same hospital
• As a very, Very, VERY sick joke (do not try this
  at home!), the hospital staff returns the babies
  to the mothers at random
• How likely is it that everyone will get the right
  baby? Or nobody will? Or at least one will?
• What is the average number that would get the
  right baby in the long run?

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Matching problem

• Simulate
• Put babys name on each of four index cards
• Shuffle cards thoroughly
• Random distribute cards to four mothers
• Repeat
• Repeat
• Repeat
• Approximate probabilities by proportions

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Matching problem

• Simulate
• Turn to technology Random Babies applet
  (www.rossmanchance.com/applets/)
• Which is the most likely outcome?
• How unlikely is getting 4 matches?
• Is 3 very unlikely or impossible?
• Whats the long-run average number of correct
  matches?
• What happens as you conduct more repetitions?

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Matching problem
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Matching problem
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Matching problem
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Matching problem
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Matching problem

• Enumeration
• 1234 1243 1324 1342 1423 1432
• 2134 2143 2314 2341 2413 2431
• 3124 3142 3214 3241 3412 3421
• 4123 4132 4213 4231 4312 4321
• Count of matches for each outcome

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Matching problem

• Enumeration
• 1234 1243 1324 1342 1423 1432
• 4 2 2 1 1
  2
• 2134 2143 2314 2341 2413 2431
• 2 0 1 0 0
  1
• 3124 3142 3214 3241 3412 3421
• 1 0 2 1 0
  0
• 4123 4132 4213 4231 4312 4321
• 0 1 1 2 0 0

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Matching problem

• Exact probabilities
• 0 matches 9/24 .375
• 1 match 8/24 .333
• 2 matches 6/24 .250
• 3 matches 0
• 4 matches 1/24 .042
• Expected value
• 0(9/24) 1(8/24) 2(6/24) 4(1/24) 24/24 1
• Long-run average value

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Collector problem

• Each cereal box is equally likely to contain any
  one of 3 prizes
• You want to collect the entire series of 3 prizes
• Suppose that you buy boxes one at a time
• What is the fewest that you might need to buy?
• What is the most that you might need to buy?
• How many boxes should you buy to have a 95
  chance of success?
• How many boxes do you expect to need (on
  average)?

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Collector problem

• What if there are k prizes?
• Same questions as above
• How likely are you to succeed if your strategy is
  to buy twice as many boxes as prizes?

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Collector problem

• Simulate!
• Write each prize on an index card
• Shuffle cards thoroughly
• Select one card at random, note which prize
• Repeat, until all prizes have been obtained
• Note the number of selections needed
• Repeat
• Repeat
• Estimate probability distribution (of number of
  boxes needed) by distribution of results

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Collector problem

• 3 prizes
• 100,000 repetitions
• Average 5.483 boxes
• Probability of success with 6 boxes 0.742
• Boxes needed for .95 probability 11

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Collector problem

• 4 prizes
• 100,000 repetitions
• Average 8.326 boxes
• Probability of success with 8 boxes 0.625
• Boxes needed for .95 probability 16

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Collector problem

• 10 prizes
• 100,000 repetitions
• Average 29.322 boxes
• Probability of success with 20 boxes 0.214
• Boxes needed for .95 probability 51

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Secretary problem

• My all-time favorite probability/math problem!
• Your task is to hire a new employee, subject to
  these constraints
• You know how many candidates have applied (n)
• The candidates arrive in random order
• You interview candidates one at a time
• You can rank candidates after interviewing them
• But you have no prior sense of quality

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Secretary problem

• More constraints
• After you have interviewed a candidate, you must
  decide immediately whether to hire
• No follow-up interviews
• No going back later
• Your task is to choose the best!
• Any other result youve failed!

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Secretary problem

• Predict the optimal probability of success when
• n 3
• n 12
• n 500
• n 4,484,451 (population of NZ)
• n 7,182,483,662 (world population)

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Secretary problem

• Enumeration for n 1
• A 1
• Probability of success 1!

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Secretary problem

• Enumeration for n 2
• A 12 B 21
• Two strategies
• Hire first candidate
• Hire second candidate
• Probability of success .5

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Secretary problem

• Enumeration for n 3
• A 123 B 132 C 213
• D 231 E 312 F 321
• Seems like probability of success 1/3
• Can we do better?

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Secretary problem

• Key insight
• We can learn from the first candidate
• Optimal strategy
• Let one go by
• Then hire first candidate who is the best so far
• A 123 B 132 C 213
• D 231 E 312 F 321

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Secretary problem

• Enumeration for n 4
• A 1234 B 1243 C 1324 D 1342
• E 1423 F 1432 G 2134 H 2143
• I 2314 J 2341 K 2413 L 2431
• M 3124 N 3142 O 3214 P 3241
• Q 3412 R 3421 S 4123 T 4132
• U 4213 V 4231 W 4312 X 4321
• Same form for optimal strategy
• But should we let 1 go by or let 2 go by?

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Secretary problem

• Let 1 go by
• A 1234 B 1243 C 1324 D 1342
• E 1423 F 1432 G 2134 H 2143
• I 2314 J 2341 K 2413 L 2431
• M 3124 N 3142 O 3214 P 3241
• Q 3412 R 3421 S 4123 T 4132
• U 4213 V 4231 W 4312 X 4321
• Probability of success 11/24 .4583

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Secretary problem

• Let 2 go by
• A 1234 B 1243 C 1324 D 1342
• E 1423 F 1432 G 2134 H 2143
• I 2314 J 2341 K 2413 L 2431
• M 3124 N 3142 O 3214 P 3241
• Q 3412 R 3421 S 4123 T 4132
• U 4213 V 4231 W 4312 X 4321
• Probability of success 10/24 .4167

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Secretary problem

• n 4
• Optimal strategy
• Let 1 go by, then hire first one who is best so
  far
• Probability of success 11/24 .4583
• Decreasing, but not as quickly as most expect

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Secretary problem
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Secretary problem
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Secretary problem

• Recall
• For large values of n
• And so

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Secretary problem

• Recall
• Goal choose r to maximize f(r)
• Elementary calculus gives
• Optimal probability of success becomes

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Secretary problem

• Remarkable result
• As n gets infinitely large
• Optimal strategy is to let first 1/e (about
  37) go by
• Then hire first who is best so far
• Optimal probability of success ? 1/e, about 37

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Secretary problem

• Extensions
• Hidden number game
• Finding your soul-mate in life

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Thanks!

• arossman_at_calpoly.edu