Question

assume that 2 cards are drawn from a standard 52 - card deck. find the following probabilities.
a) assume the cards are drawn without replacement. find the probability of drawing a spade and a heart.
b) assume the cards are drawn with replacement. find the probability of drawing a spade and a heart.
a. the probability of drawing a spade and a heart, without replacement, is \\( \frac { 13 } { 102 } \\).
(simplify your answer.)
b. the probability of drawing a spade and a heart, with replacement, is \\( \square \\).
(simplify your answer.)

Explanation:

Step1: Calculate the probability of each event with replacement

In a standard 52 - card deck, there are 13 spades and 13 hearts.
The probability of drawing a spade on the first draw is \(P(S)=\frac{13}{52}\).
Since the card is replaced, the probability of drawing a heart on the second draw is \(P(H)=\frac{13}{52}\).
Also, the probability of drawing a heart on the first draw and a spade on the second draw is \(P(H_1)=\frac{13}{52}\) and \(P(S_2)=\frac{13}{52}\)

Step2: Use the addition rule for mutually - exclusive events (since \(S\) then \(H\) and \(H\) then \(S\) are two different non - overlapping cases)

The probability of drawing a spade and a heart (in either order) with replacement is \(P=( \frac{13}{52}\times\frac{13}{52})+( \frac{13}{52}\times\frac{13}{52})\)

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Answer:

\(\frac{1}{8}\)