Random Variables

Question

In a card game, a player randomly selects 2 cards in succession from a 52-card bridge deck. The player receives a score of 10 if the second card is the same suit as the first, a score of 5 if the second card is the same color and different suit, and a score of 0 if the second card is a different color. Let the random variable X be the score and find the probability distribution of X. (Enter your probabilities as fractions.)
X         P(X)
0
5
10

Answer 1

No matter what the first card is, there are 3 possible outcomes from the second card:

 

Same suit as the first -- 10 points -- after the first card is drawn, there are 51 cards left in the deck, of which 12 are the same suit as the first.  So the probability of this outcome is 12 / 51

 

Different suit, same color -- 5 points -- after the first card is drawn, there are 51 cards left in the deck, of which 13 are the other suit of the same color as the first.  So the probability is 13 / 51

 

Different color -- 0 points -- after the first card is drawn, there are 51 cards left in the deck, of which 26 are one of the suits with the other color.  So the probability is 26 / 51

 

So the probability distribution is:

 

X         P(X)

---       -------
0         26/51
5         13/51
10        12/51

 

The total, as it should be, is 51/51 = 1.