Two cards are drawn at random from a shuffled deck.

1. What is the probability that at least one is a heart?

(b) If you know that at least one is a heart, what is the probability that both are

hearts?

There are two cards picked from a deck of 52 cards.

The events are said to be independent when the occurrence or non-occurrence of any event does not have any effect on the occurrence or non-occurrence of the other event.

Probability of at least one heart can be found by subtracting the probability of no heart from 1.

For the first card to be a non-heart, there are 39 choices out of 52 and the second card is 38 out of 51 cards as there is no replacement, this implies that $\mathit{P}\left(noheart\right)\mathbf{=}\frac{\mathbf{39}}{\mathbf{52}}\mathbf{\times }\frac{\mathbf{38}}{\mathbf{51}}$

Find the probability at least one heart is picked.

$$\begin{gathered} P\left(atleast1heart\right)=1-\frac{39}{52}\times \frac{38}{51} \\ =1-\frac{19}{34} \\ =\frac{15}{34} \end{gathered}$$

The probability of both hearts is $P\left(bothheart\right)=\frac{13}{52}\times \frac{12}{51}$

Find the probability both are heart when at least one is heart using the Bayes theorem, $P_A\left(B\right)=\frac{P\left(AB\right)}{P\left(A\right)}$.

$\mathit{P}\left(bothheartwhenatleastoneheart\right)=\frac{{\displaystyle \frac{13}{52}}\times {\displaystyle \frac{12}{51}}}{15}$

Thus, the desired probability is $\frac{2}{15}$.