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I have the following problem:
You have a standard deck of 52 cards.
The cards: 2,3,4,5,6,7,8,9,10,J,Q,K,A
J, Q, K, A have the values 11, 12, 13, 14 appropriately.

Two cards are drawn from the deck without replacement.
Let X be the absolute difference between the 2 cards.

Find the probability mass function of X?

What I've solved so far:
P(X=0) = 3/51 (reasoning: when you draw a card the second time, there are 3 cards left with the same value as the first one)
P(X=12) = (4/52 * 4/51) + (4/52 * 4/51) = 2 * (4/52 * 4/51) (reasoning: to have a difference of 12 you need to draw a 2 the first time, and an A the second time, or the other way around)

Please help me determine the rest (X=1...13)

Ben Watson
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1 Answers1

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First think about the range that the random variable $X$ can take. Note $X$ can take $0,1,2,...,12$.

$X = 0:$ You pick two cards of the same denomination without replacement. The probability of choosing them is $(4/52)*(3/51)$ since the drawings are independent and without replacement. This picking can occur for any one of the $13$ denominations. So, $P[X = 0] = 13*(4/52)*(3/51) = 3/51$

For $X = i$, when $i>0$: note that you perform a similar action, i.e., drawing a denomination at random from $52$ cards followed by a second draw of another denomination different from the first one out of the remaining $51$ cards. So, the probability of this event is $(4/52)*(4/51)$.

But note that for different mass points of X, the choices of two denominations will vary. Think of two cases among the choices: $1.$ The first draw is always a smaller denomination than the second draw, $2.$ The second draw is always a larger denomination than the first one.

For $1.$ with $X = 1$, the number of cases is $(13-1) = 12, viz.,\{(2,3), (3,4), .., (13,14)\}$, with $X = 2$, the number of cases is $(13-2) = 11, viz.,\{(2,4), (3,5), .., (12,14)\}$ and so on until $X = 12$ (i.e., when $\{(2,14)\}$)

For $2.$, it is similar to $1.$ [only the order is opposite]. So, you have $$P(X = 0) = 3/51,$$ And, $$P(X = i) = 2*(13 - i)*(4/52)*(4/51), i=1(1)12.$$

Sauvik De
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