ENGRD2700_recitation5_solutions - ENGRD2700 Basic...

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ENGRD2700 Basic Engineering Probability and Statistics Fall 2011 Recitation 5: 23-26 September 2011 1. Roulette is one of the most popular casino games played across the world. In this problem we will examine the probabilities associated with this “game of chance”. The game consists of spinning a wheel containing slots for the numbers 1 to 36, and the numbers 0 and 00. Each of the numbers 1 to 36 is coloured either red or black (with exactly 18 red numbers and 18 black numbers), and the other two are coloured green. A ball is tossed around the wheel and eventually lands in one of the slots, selecting that number. It is assumed (and should be practically the case if the wheel is properly manufactured) that the outcome of one spin has no impact on the outcome of any other spin, i.e. that spins are independent, and that on each spin, every number is equally likely to be selected. Gamblers play by betting on the outcome of each spin, by placing chips on a layout representing the possible outcomes (see last page). Bets can be made on individual numbers or on certain combinations, such as strings of consecutive numbers, low or high numbers, or the colour of the number. Bets on individual numbers pay 35 to 1, so if $1 is bet on, say, 12, and that number comes up, the gambler keeps the $1 and wins an additional $35. Bets on combinations of numbers are paid in such a way that you win the same betting on the combination as you would if you bet on each of the numbers in the combination individually, with the same total bet amount. For example, a bet of $1 on { 17 , 18 } wins $17, which is the same amount you would win if you bet 50 cents on both 17 and 18. Similarly, a bet on Red pays 1 to 1, which is the same payoff you would see as if you bet individually on each of the 18 red numbers. (a) If you bet $1 on a combination bet consisting of n numbers, how much will you get paid if you win (find a formula in terms of n )? (36 - n ) /n (b) Let the random variable X denote the outcome of one spin. X will take one of the values { 1 , 2 ,..., 36 , 0 , 00 } . What is the probability P ( X = 12)? What is the probability P ( X = x ) for any of the possible outcomes x ? What (named) type of discrete distribution does this correspond
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ENGRD2700_recitation5_solutions - ENGRD2700 Basic...

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