Probability Take-Home Exam Questions

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$Math 232 syllabus - Denison University$

$Math 102 syllabus - Personal Pages - Denison University$

S1S2S3 S4 S5 S6 S7S8S9R1 R2 R3 R4 R5 R6 R7R8R9Figure 1: A Plinko Board(b) What is the expected value of your Plinko chip (for the optimal slot)?10. Tournament of Luck (30 pts)Each year, very lucky people from all over the world assemble in Las Vegasto determine who is the luckiest of all. Assume there are 8 players inthe tournament. The players are initially randomly paired in a tournamentbracket as shown below:1. ----2. ----3. ----4. ----5. ----6. ----7. ----8. -------------------------------- ( tournament winner )For each match the referee flips a coin to determine the winner. The losers4
are eliminated and the winners continue to the next round to compete againstother winners.(a) If two sisters are among the eight entrants, what is the probability thatthey compete against each other at some point during the tournament?(b) Instead of 8 players, assume there are 2 n players for some value of n.Find a formula for the probability that the two sisters will meet eachother in this tournament.(c) Use your formula from part (b) to find the probability that two sistersmeet up somewhere in a contest with 1024 people.11. Birthday Holidays (35 pts)Game theory class is so interesting that they decide to meet every single dayof the year. Every day they meet, each person in the class works on and thensolves a new problem. However, if anyone in the class has a birthday onthat particular day, they all take a break and have a party instead. How manypeople should be permitted in the game theory class if the professor wants tomaximize the total expected number of newly solved problems for the wholeyear?Note this problem requires basic calculus skills.12. Price is Right Wheel (35 pts)The Price is Right television game show has a big spin-off wheel to see whogets in the show-case show down. Three contestants take turns spinning thewheel which have 20 spaces with amounts of $0.05 to $1.00 in five-cent increments.Each contestant is allowed one or two spins where the goal is tohave the largest total amount that does not exceed $1.The first person must often decide whether they want to spin again and riskgoing over a dollar, or trying to get a higher total sum. What is the optimalstrategy for the first player (i.e., what dollar amount should they stay versusspin again)? Answer this problem for only two contestants (it is easier thanconsidering all three).13. Two Urns (35 pts)There are two urns. Urn A contains 2 red balls and 1 black ball. Urn Bcontains 101 red balls and 100 black balls. You do not know which urn iswhich. Your goal is to select two balls from the same urn and then guess5
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Probability Take-Home Exam Questions

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