Question

One of the more interesting applications of the x test is to examine issues of fairness and cheating in games of chance. Since such games usually involve wagering, there is an incentive for people to try to rig the games, and allegations of missing cards, "loaded" dice, and "sticky" roulette wheels are all too common. Take the example of dice. Most dice used in wagering have six sides, with each side having a value of one, two, three, four, five, or six. If the die being used is fair, then the chance of any particular number coming up is the same: 1 in 6. However, if the die is loaded, then certain numbers will have a greater likelihood of appearing, while others will have a lower likelihood. 2. After Turner finishes at the Blackjack table, you (the casino's statistician) follow him to a craps table. In two hours of playing, he's racked up $30,000 in winnings and is showing no sign of stopping. Crowds are gathering around him to watch his streak, and he is telling everyone within earshot that his good luck is due to the fact that he's using the casino's lucky pair of "bruiser dice," so named because one is black and the other blue. Unbeknownst Turner, however, you have been quietly watching his rolls and marking down the values of each roll, noting the values of the black and blue dice separately. After 60 rolls, you have become convinced that the blue die is loaded: you've recorded the blue die rolling a one 16-times; a two 5- times; a three 9-times, a four 7-times; a five 6-times, and a six 17-times. The x test can be used to estimate the likelihood that the values observed on the blue die occurred by chance. Craps data Dice Rolls 1. It's Friday night at the Tunisian Nights Casino, and renowned gambler Jeremy Turner is having a fantastic night at the blackjack table. Out of all of the 36 games played, he beats the dealer 28 times and loses to the dealer 8 times. You are convinced that he has been counting cards. The x- test can be used to estimate the likelihood that the number of wins Turner has had occurred by chance. Note: the total for both the observed and expected will be the total number of games he has played. a=.05 Total Observed Expected Blackjack data Game result Total a= .05 df = critical x = x2= Significant? Y or N Observed Expected Interpret the results of the y analysis. df = critical x2 = x2 = Significant? Y or N Interpret the results of the x_ analysis. 3. Based on the results of the ye analyses, would you inform the owner of the casino of your discoveries and recommend Turner be thrown out? In other words, is Turner cheating? Why/why not?

Answer 1

CHI-SQUARE GOODNESS of fit TEST: Test Statistic given by X² E CA-12 ei cohere Obseared frequency ei. Expected Expected frequen chi-square distribution frequently follows with (n-1) df. Now

00 Number wins doesnot Hypothest Hypothesis (Ho) of Chance. wins Alternative Hypothesis (fi) typothest (his: Number of haz ocurre Chonce , by

Under Mall Hypothesis Expected freque uency ei = 28+ 8 - 36 36 18 Game Result win lose total 28 oo BI 28 ei 18 18 28

fi ei (ti-ei) (fi-eil 2 (fi-ei) lei 28 18 100 S.56 18 -lo 8 10o 5.56 NOW, ya 5.50 +5.56147 Y llein de nel 2-1 21 Now Critical value for (A.D. = I degree of freedom and 0.05 level of significance is critical Value 3.841 Since Critical Value L ta calculated. so Our null at 0.05. may dreject trypothests level significance Significant There fore It is ) Number wins Turnen we conclude that the has had ocawired of Chance. by