Probability Question - None

Question

A, B, C and D cut a pack of 52 cards successively in the order given. If the person who cuts a spade first receives ₹350, then find the expectations of:

(a) B is ₹96

(b) D is ₹54

(c) (A + C) is ₹200

(d) (B - D) is ₹56

Accepted Answer

Solution:

Let E be the event of any one cutting a spade in one cut and let S be the sample space.

Then $n(E) = \binom{13}{1} = 13$ and $n(S) = \binom{52}{1} = 52$

$p = P(E) = \frac{n(E)}{n(S)} = \frac{13}{52} = \frac{1}{4}$

$q = P(E') = 1 - p = \frac{3}{4}$

The probability of A winning (when A starts the game):

$$P(A) = p + q^4p + q^8p + \ldots = \frac{p}{1-q^4} = \frac{1/4}{1-(3/4)^4}$$

Expected value of A's winnings: $E(A) = 350 imes P(A)$

Expected value of B's winnings: $E(B) = 350 imes qp(1 + q^4 + q^8 + \ldots) = ₹96$

Expected value of D's winnings: $E(D) = 350 imes q^3p(1 + q^4 + q^8 + \ldots) = ₹54$

∴ Answers are (a), (b), and (c).

Question:

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