1 Econ 310 Professor Wallace Fall 2013 Week 7 Lab Handout Answers Keno. Keno is a popular casino game. To play the game gamblers bet on selection of numbers 1 through 80. Then, from a group of 80 balls labeled 1 through 80, 20 balls are drawn at random and without replacement. Payouts for Keno are based on how many numbers the gambler chooses to bet and how many balls are drawn with the gambler’s numbers. One popular bet is for the gambler to bet on 2 numbers. At Caesars Palace a gambler placing this type of bet will win $12 for each dollar wagered if both the gambler’s numbers are selected in the ball draw and nothing if fewer than 2 of the numbers are selected. Thus, if a gambler bets $10 on X and Y he would win $120 if both the numbers X and Y were selected, and would lose $10 otherwise (if either number X or Y or both were not selected). A) Calculate the probability of winning on a single Keno bet on 7 and 13. What are the expected winnings in a keno game when betting $10 on numbers 7 and 13? This is just the hypergeometric probability of 2 successes in 20 trails when selecting 20 objects from of population of 80 with 2 successes in the population 2 782 1820 19 19( ) 0.06018080 79 31620P Win          19 if $120 316()297 if 0 316iiiWfWW 19 297( ) 120 10 2.18316 316WiEW              B) What is the variance of winnings in a keno game when betting $10 on numbers 7 and 13?    22219 297=Var( ) 120 ( 2.18) 10 ( 2.18) 955.04316 316WiW                2 Another way to calculate the expected value and variance is to note that the winnings associated with the i’th keno game is 130 10iiWX where 1iX  if Jake won the i’th game and 0iX  if Jake lost the i’th game. We know that 191,316Xb so using our properties of expected value and variance of linear transformations 2219( ) 130 10 2.1831619 19( ) 130 1 955.05316 316WiWiEWVar W              *** Note there are slight differences in the variance calculated different ways due to rounding. C) Your friend Jake plans visit Caesar’s Palace where he will play 200 keno games, betting $10 on his lucky numbers 7 and 13 in each game. What is the approximate distribution of Jake’s average winnings.1 A complete answer will describe the distributional form, the mean, and the variance Jake’s average winnings. 2955.04, 2.18,200WWW N W Nn D) Jake returns from his trip, informs you that he won $210, and won’t stop talking about how playing keno with his “system” of betting 7 and 13 keno is easy money. How lucky did your friend Jake get? (Hint: What is the probability that Jake would win less than $210?). If Jake wins $210 his average winnings are  111210 1.05200niiwwn   We know that his result puts him somewhere in the right hand tail of the distribution of average winnings. To determine how lucky he got we need to figure out how far in the right hand tail his result is. We can do this by calculating the probability of having average winnings more or winning less than 2.05  1.05 ( 2.18)1.05 1.48 0.0694955.04200P W P Z z      1 Jake’s average winnings are given by 11niiWWn where iW are his winnings on the i’th keno game and n the number of games he plays.3 Thus, about 7% of the time Jake would win more than an average of 1.05 and about 93% of the time he would win less than an average of 1.05. Assuming there is nothing to his system it appears that he got pretty lucky on his trip to Vegas. E) Jake is hooked on keno and decides to start playing in a local illegal gambling club run by organized crime figures. After playing 200 keno games in the local gambling club Jake loses $570 and claims the local keno game is rigged against his system. i. Assuming that local keno game is fair, how likely is it that Jake would lose $570 or more? For Jake’s play in the local keno club  111570 2.85200niiwwn      2.85 ( 2.18)2.85 0.31 0.3783955.04200P W P Z z          ii. Does this probability suggest anything about the fairness of the local Keno game? Jakes result was not atypical for a fair keno game. About 38% of the time Jake playes 200 fair keno games betting $10 on 2 numbers in each game he would lose more than an average of $2.85. Thus, his results certainly doesn’t suggest that the keno game is rigged. $ This probability is the p-value associated with the test of the null hypothesis 0: 2.18WH that we talked about in class on October 22. Because the p-value is larger than any reasonable significance level we cannot reject the null that the keno game is fair or better for Jake. F) Repeat parts C through F, this time using information about the distribution of the sample winning proportion instead of average winnings (i.e., the distribution fraction of times Jake wins in 200 plays as opposed to his average winnings after 200 plays). Be sure to clearly state the distributed of the proportion of wins at the outset (i.e., the equivalent to part C.). (Tip: For this problem you will need to calculate the number of wins out of 200 plays that corresponds with winning $210 and losing $570, use these number of wins to calculate the proportion of wins for Jake’s trip and play at the local club, and then calculate the probability of winning a lesser proportion of times). C) 19 297(1 ) 19316 316,,316 200ppP N p P Nn4 D) Note that 130 2000WX   where X is the number wins in 200 plays betting $10 in each play. Thus, winning 210 means that Jake won 17 times out of 200 plays for a (sample) winning proportion of 170.085200p .2 We know that his result puts him somewhere in the right hand tail of the distribution of the proportion of winnings in 200 plays. To determine how lucky he got we need to figure out how far in the right hand tail his result is. We can do this by