Q. 2.36 - A First Course in Probability

Question

Two cards are chosen at random from a deck of $52$playing cards. What is the probability that they

(a) are both aces?

(b) have the same value?

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right)0.0045 \\ \left(b\right)0.059 \end{gathered}$$

Step 1 Given Information.

Two cards are chosen at random from a deck of $52$playing cards.

Step 2 Part (a) Explanation.

$$\begin{gathered} \text{P((both aces)=P(first ace)}{}^{*}P(second ace) \\ =(4 / 52) *(3 / 51) \\ =0.0045 \end{gathered}$$

Step 3 Part (b) Explanation.

$$\begin{gathered} P\left(\right(both same value)=[Choose a value\left]{}^{*}P\right(first card\left){}^{*}P\right( second card with same value) \\ =(13C1)*(4/52)*(3/51) \\ =13 *(1 / 13) *(1 / 17) \\ =0.059 \end{gathered}$$