At a certain point in a card game, if you get spade, you win $4, if you get a 2

Question

At a certain point in a card game, if you get spade, you win $4, if you get a 2 (except the 2), you lose $5, if you get the A, you win $20, and if you get any other card, the game ends with no money being exchanged.

a) What is your expected gain or loss?

b) You and 3 friends are playing using the rules above each using your own deck. We are interested if a player receives any of the cards listed.

i) What is the expected value of the number of people getting one of the above cards?

ii) What is the variance of the number of people getting one of the above cards?

Explanation / Answer

a) P(spade) = 13/52 = 1/4

P(2, except the 2) = 3/52

P(A) = 1/52

Expected gain = sum of (probability x corresponding gain or loss)

= (1/4) x 4 + (3/52) x -5 + (1/52)x20 = $1.096

So, the expected gain is $1.096

b) This can be considered as binomial distribution, with n = 3+1 =4

p = P(getting one of the above cards) = (1/4) + (3/52) + (1/52) = 17/52

q = 1 - (17/52) = 35/52

i) Expected value of the number of people getting one of the above cards = np = 4x17/52 = 1.31

ii) Variance of the number of people getting one of the above cards = npq = 4x(17/52)x(35/52) = 0.88

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