When playing the game two-up, two coins are tossed in the air, and there are thr

Question

When playing the game two-up, two coins are tossed in the air, and there are three ways that they can land: heads (that is, both coins land heads), tails, or odds (that is, one heads and one tails). In each game, the probability of heads is 1/4, the probability of tails is 1/4, and the probability of odds is 1/2. The result from different games of two-up can be assumed to be independent (a) When betting on a game of two-up, the house wins all bets if the game starts with the coins landing odds five times in a row. What is the chance of this happening? (b) Let X be the amount you win from a round of two-up, according to the above rules, when you bet S1. The following is the probability distribution of the amount you will win: Value ofX 02 17 15 32 32 Probability (i) Is X a continuous or discrete random variable? (ii) Find the expected value of X. (iii) On average, do you expect to win more from a round of two-up than it costs to play, in the long run? (iv) Find the standard deviation of X.

Explanation / Answer

(a) Pr(Coin landing odds five times in row)= Pr(Coin landing odds one time)5 = (1/2)5 = 1/32

(b)(i) Here X is discreter random variable.

(ii) E(X) = 0 * 17/32 + 5 * 15/32 = 30/32 = 15/16

(iii) No we don not expect to win more from a round of two up than it costs to play as cost is $1.

(iv) Var(X) = 17/32 * (0 - 30/32)2 + 15/32 * ( 2 - 30/32)2 = 255/256

Std(X) = 0.998

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