Answer to Question #336687 in Statistics and Probability for Ohemaa lizzy

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. Consider two possibilities:

1. One card is drawn. There are $52$ cards and $8$ 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 $0.02$(it is rounded to two decimal places) for two cards. The probability to get $4$ is $\frac{1}{13}$. The probability to get not a heart is $\frac{3}{4}$.