5 cards are drawn at random from a standard deck. find the probability that all the cards are hearts. find…

Explanation:

Step1: Calculate total number of ways to draw 5 - card hands

The total number of 5 - card hands from a standard deck of 52 cards is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 52$ and $r=5$. So, $C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960$.

Step2: Calculate number of ways to draw 5 - heart hands

There are 13 hearts in a deck. The number of 5 - heart hands is $C(13,5)=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287$. The probability that all 5 cards are hearts is $\frac{C(13,5)}{C(52,5)}=\frac{1287}{2598960}=\frac{33}{66640}\approx0.000495$.

Step3: Calculate number of face - card hands

There are 12 face - cards in a deck (4 kings, 4 queens, 4 jacks). The number of 5 - face - card hands is $C(12,5)=\frac{12!}{5!(12 - 5)!}=\frac{12\times11\times10\times9\times8}{5\times4\times3\times2\times1}=792$. The probability that all 5 cards are face - cards is $\frac{C(12,5)}{C(52,5)}=\frac{792}{2598960}=\frac{33}{108290}\approx0.000305$.

Step4: Calculate number of even - card hands

The even - numbered cards are 2, 4, 6, 8, 10. There are 5 ranks of even - numbered cards and 4 suits for each rank, so there are $5\times4 = 20$ even - numbered cards. The number of 5 - even - card hands is $C(20,5)=\frac{20!}{5!(20 - 5)!}=\frac{20\times19\times18\times17\times16}{5\times4\times3\times2\times1}=15504$. The probability that all 5 cards are even is $\frac{C(20,5)}{C(52,5)}=\frac{15504}{2598960}=\frac{323}{54145}\approx0.00597$.

Answer:

Probability that all cards are hearts: $\frac{33}{66640}$ Probability that all cards are face - cards: $\frac{33}{108290}$ Probability that all cards are even: $\frac{323}{54145}$