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 2 black cards.
b) assume the cards are drawn with replacement. find the probability of drawing 2 black cards.
a. the probability of drawing 2 black cards without replacement is \\(\frac{25}{102}\\)
(simplify your answer.)
b. the probability of drawing 2 black cards with replacement is \\(\square\\)
(simplify your answer.)

Explanation:

Step1: Calculate the probability of drawing a black card in the first draw

In a standard 52 - card deck, there are 26 black cards. The probability of drawing a black card in the first draw with replacement is \(P_1=\frac{26}{52}=\frac{1}{2}\).

Step2: Calculate the probability of drawing a black card in the second draw

Since the card is replaced, the deck composition remains the same. The probability of drawing a black card in the second draw is \(P_2 = \frac{26}{52}=\frac{1}{2}\).

Step3: Use the multiplication rule for independent events

For independent events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\). Here, \(A\) is the event of drawing a black card in the first draw and \(B\) is the event of drawing a black card in the second draw. So \(P = P_1\times P_2=\frac{1}{2}\times\frac{1}{2}\).

Answer:

\(\frac{1}{4}\)