Find the probability of the following five-card poker hands from a 52-card deck. In poker, aces are either high or low.
-Four of a kind ( 4 cards of the same value)
A) $\frac { 1 } { 4165 }$
B) $\frac { 4 } { 4165 }$
C) $\frac { 1 } { 4080 }$
D) $\frac { 13 } { 49980 }$

Question

Quizplus · Accepted Answer

To get four of a kind, you first choose the rank of the cards that will be the four of a kind (13 options), and then choose 4 out of the 4 cards of that rank (which is only 1 way). For the fifth card, you have 48 options (52 cards in the deck minus the 4 cards of the chosen rank). So, the total number of ways to get a four of a kind is $13 \times 1 \times 48 = 624$. The total number of 5-card hands from a 52-card deck is $\binom{52}{5} = 2,598,960$. Therefore, the probability is $\frac{624}{2,598,960} = \frac{1}{4165}$.

Find the Probability of the Following Five-Card Poker Hands from a 52-Card