Construct a probability mass function of all possible outcomes and find the expected amount that the gambler will get.

Question

In a distant past where gambling is still allowed, a gambling King Pin (who is really good in math), placed a very tricky game. Given one deck of cards, a gambler (who is not good in math) is allowed to select one card from each suit (diamond, spade, heart, clubs) and stake the same amount on each chosen card. Now, shuffling each suit separately, the gambler will select a card from each suit. If the selected card appears once, twice, thrice or four times, then the gambler will get money one, two, three or four times of the stake respectively, and the original stake also returned to him. However, if the selected card does not appear, he will lose his stake.

Answer 1

For simplicity, let the stake be $1. The possible winnings are -$1, $1, $2, $3, and $4. The way to find the expected winnings is to multiply each amount by the probability that it occurs, and then add up the 5 results. Since there are 13 cards in a suit, the odds of a match for a single suit are 1 in 13.

This problem fits the criteria for a binomial distribution. N (the number of trials) is 4, p (the probability of the desired outcome) is 1/13, and X (the number of successes) is 0, 1, 2, 3, or 4.

To find each of the 5 probabilities, use the binomial probability formula P(x) = _NC_x·p^x(1-p)^(N-x)

Alternatively, you can use a graphing calculator to input N, p, and x into the binompdf function.

You should get the following results (rounded to the nearest .001):

1. P(0) = 0.726
2. P(1) = 0.242
3. P(2) = 0.030
4. P(3) = 0.002
5. P(4) = 0.000

Multiplying each of these by its value and adding gives you -.726+.242+.060+.006+.000 = $-0.418