Question

you are to pick 4 cards randomly from a condensed deck of cards that contains four suits - ♡,◇,♧,♤ - and the following denominations: ace, 2, 3, 4, 5, 6, 7, 8, 9 and 10. there are no face - cards in this deck. what is the probability that you will get part (a) a three - of - a - kind (three of the same denomation)? p(3 of a kind)=0.0158 (use four decimals) part (b) what is the probability that all four cards are of the same suit? for example, all four cards are ♡s? p(all the same suit)=0.0092 (use four decimals) part (c) what is the probability you get one - pair? p(one pair)=0.3782 (use four decimals) part (d) what is the probability you get two aces and two 10s? p(2 - aces and 2 - 10s)=0.0004 (use four decimals) part (e) what is the probability you get two aces and two ♡s? p(2 - aces and 2 - ♡)= (use four decimals)

Explanation:

Step1: Calculate total number of ways to pick 4 cards

The deck has $4\times9 = 36$ cards. The number of ways to choose 4 cards out of 36 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 36$ and $r=4$. So, $C(36,4)=\frac{36!}{4!(36 - 4)!}=\frac{36\times35\times34\times33}{4\times3\times2\times1}=58905$.

Step2: Calculate number of ways to get two aces and two hearts

There are 4 aces and 9 hearts in the deck. But we need to be careful not to double - count. The number of ways to get 2 aces out of 4 is $C(4,2)=\frac{4!}{2!(4 - 2)!}=\frac{4\times3}{2\times1}=6$. The number of non - ace hearts is $9 - 4=5$. The number of ways to get 2 non - ace hearts out of 5 is $C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10$. So the number of ways to get 2 aces and 2 hearts is $C(4,2)\times C(5,2)=6\times10 = 60$.

Step3: Calculate the probability

The probability $P$ is the number of favorable outcomes divided by the number of total outcomes. So $P=\frac{60}{58905}\approx0.0010$.

Answer:

$0.0010$