Question

Number 38 please.

Secuon A./ 37. Suppose we have the lottery in Exercise 30. Assume all possible tickets are printed and all tickets are distinct. (a) Compute the probability of winning if one ticket is purchased. (b) Compute the probability of winning if seven tickets are purchased. 38. Suppose a poker hand (five cards) is dealt from an ordinary deck (52 cards) (a) Compute the probability of the hand containing four aces. (b) Compute the probability of the hand containing four of a kind. 39. Suppose a fair six-sided die (that is, the probability of each side turning up is b) is to be rolled. The player will receive an amount of dollars equal to the number of dots that turn up, except when five or six dots turn up, in which case the player will lose $5 or $6, respectively. (a) Compute the expected value of the amount of money the player

Answer 1

Answer 38

(a) 48/(⁵²C₅)

Number of ways in which we can choose 5 cards out of 52 cards( without repeating any card) is given by ⁵²C₅ = total number of possible outcomes.

The fifth card ( any card other than the ace) can be any of the remaining 48(52-4) cards. Therefore the number of favourable outcomes are 48 ( 4 cases plus anyone of the 48 other cards).

Therefore the required probability = 48/(⁵²C₅).

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(b) (1/49)x(2/50)x(3/51)

Having 4 of a kind means 4 of same the same rank. We can choose any one of the 13 ranks in 52 ways out of tatal 52 ways, hence probability of 52/52. Once this is chosen, we can select the next(second) of same rank in 3 ways out of 51 , hence probability 3/51, then next(3rd ) of same rank with probability of 2/50 , finally the fourth of the same rank with probability 1/49, and the last one can be any one of the remaining 48 cards with probability 48/48.

Hence the final probability is 52/52 x 3/52 x 2/50 x 1/49x 48/48 = 1/49 x 2/50 x 3/51.

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