Answer to Question #337300 in Statistics and Probability for Manuel frost

Question

A deck (52) of cards (clubs, diamonds, hearts and spades) is shuffled once. Estimate the probability of drawing a king and queen. Estimate the probability of drawing a 4 (four). Estimate the probability of drawing a card that is not a heart.

Answer 1

It is not clear how many cards are drawn at first time. Therefore, we consider two possible cases:

1. One card is drawn. The deck contains $8$ kings and queens. Thus, the probability is: $p=\frac{8}{52}=\frac{2}{13}$
2. Two cards are drawn. There are $C_{52}^2=\frac{52\cdot51}{2}=1326$ different combinations. There are $C_8^2=\frac{8\cdot7}2=28$ combinations that contain a king and a queen. The probability to get a king and a queen is: $\frac{28}{1326}\approx0.02$. (it is rounded to 2 decimal places)

For all other cases we assume that there is only one card. The probability of getting $4$ is: $\frac{4}{52}=\frac{1}{13}$. There are $52-13=39$ cards that are not heart. The probability to get not a heart is: $\frac{39}{52}=\frac{3}{4}$.

Answer: The probability to get a king and a queen is: $\frac{2}{13}$ for one card and $0.02$ for two cards (it is rounded to two decimal places). The probability to get $4$ is: $\frac{1}{13}$. The probability to get not a heart is: $\frac{3}{4}$