You are given 12 to 1 odds against drawing two hearts when two cards are selected at random from a standard deck of 52 cards (with replacement of the first card before the second card is drawn). This means that you win $12 if you succeed and you lose $1 if you fail. Find the expected value (to you) of the game. Round to the nearest cent
To calculate the expected value of the game, we need to find the probability of winning and the probability of losing.
The probability of drawing two hearts when two cards are selected is the product of the probability of drawing a heart on the first draw and the probability of drawing a heart on the second draw. There are 13 hearts in a deck of 52 cards, so the probability of drawing a heart on any specific draw is 13/52.
Since the first card is replaced before the second card is drawn, the probability of drawing a heart on the second draw is also 13/52.
Therefore, the probability of drawing two hearts is (13/52) * (13/52) = 169/2704.
The probability of losing, which is the complement of winning, is (1 - 169/2704) = 2535/2704.
The expected value is calculated by multiplying each outcome by its probability and summing them:
Expected value = (12 * 169/2704) + (-1 * 2535/2704)
= 2028/2704 - 2535/2704
= -507/2704
Rounding to the nearest cent gives -0.19.
Therefore, the expected value of the game to you is -$0.19.