Solutions - 5030-04

Let $C_1$ be the outcome of the first card, $C_2$ be the outcome of the second card, and $H$ denote “hearts.”

(a)

Apply the multiplication rule for sequential draws without replacement:

$$\begin{array}{c}P[C_1=H\cap C_2=H]=P[C_1=H]\cdot P[C_2=H|C_1=H]\\ \\ \gg \gg \frac{13}{52}\cdot \frac{12}{51}\gg \gg \frac{1}{17}\end{array}$$

---

(b)

Apply the multiplication rule:

$$\begin{array}{c}P[C_1=4\cap C_2=4]=P[C_1=4]\cdot P[C_2=4|C_1=4]\\ \\ \gg \gg \frac{4}{52}\cdot \frac{3}{51}\gg \gg \frac{1}{221}\end{array}$$