Marcos is doing a card trick using a standard 52-card deck. He shows his friend a card, replaces it, and then shows his friend another card. What is the probability that the first card is a club and the second card is not a heart?

Jun 2, 2016

Probability $= \frac{3}{16}$

Explanation:

Define:
$\textcolor{w h i t e}{\text{XXX}} F :$ first card is a club.
$\textcolor{w h i t e}{\text{XXX}} S :$ second card is not a heart.

Assuming $F$ and $S$ are independent (although this is not obvious since the question talks about the second card as "another card" which might be read to imply that although the first card was put back in the deck, it is somehow eliminated from being chosen as the second card).

$P \left(F\right) = \frac{1}{4} \textcolor{w h i t e}{\text{XXX}}$since one quarter of the cards are clubs
$P \left(S\right) = \frac{3}{4} \textcolor{w h i t e}{\text{XXX}}$since three quarters of the cards are not hearts.

For independent events
color(white)("XXX")P(F&S)=P(F)xxP(S)
$\textcolor{w h i t e}{\text{XXXXXXxX}} = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16}$