Question

What is the probability of getting four cards of the same value ( 4 Aces, 4 Kings etc) from a pack of cards, without replacement?

Answer 1

Problem #6166 What is the probability of getting four cards of the same value ( 4 Aces, 4 Kings etc) from a pack of cards, without replacement? Please explain your answer Answer The space of elementary events in the problem is $\Omega = \{(A_1,A_2,A_3,A_4)\}$, where $A_{i}$, $i = 1,\ldots ,4$ is the i-th card in a taken group of 4 cards. Hence $|\Omega | = \binom{52}{4}$. The event we are interested in is $A = (A_{1},A_{2},A_{3},A_{4})$, where $A_{i}$ represent the card of the same value, there are 13 possible groups of such type, hence $|A| = 13$. The probability in problem, after simplifying, $p = \frac{|A|}{|\Omega|} = \frac{13}{\binom{52}{4}} = \frac{1}{25\cdot 17\cdot 39} \approx 4.8\cdot 10^{-5}$.

Answer $p = \frac{1}{25\cdot 17\cdot 39}\approx 4.8\cdot 10^{-5}$