Question

michelle draws a card from a standard deck of 52 cards. she replaces the card and draws a second card. what is the probability that she draws a heart and then a spade? note: there are 13 each of hearts, spades, clubs, and diamonds.
a (\frac{1}{16})
b (\frac{1}{34})
c (\frac{1}{4})
d (\frac{1}{13})

Explanation:

Step1: Calculate the probability of drawing a spade in one draw

The probability of an event \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
In a standard deck of \(52\) cards, there are \(13\) spades. So the probability of drawing a spade in one draw \(P_1=\frac{13}{52}=\frac{1}{4}\).
Since the card is replaced, the second - draw probability is the same as the first - draw probability.

Step2: Use the multiplication rule for independent events

If two events \(A\) and \(B\) are independent (in this case, the first draw and the second draw are independent because the card is replaced), then \(P(A\cap B)=P(A)\times P(B)\).
Here \(P(A) = P(B)=\frac{1}{4}\), so \(P = \frac{1}{4}\times\frac{1}{4}=\frac{1}{16}\)

Answer:

A. \(\frac{1}{16}\)