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Chapter 11: Problem 20

You draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a black card each time.

Short Answer

Expert verified
The probability of drawing a black card twice with replacement is 0.25.

Step by step solution

01

Determine the probability of drawing a black card in one draw

The total number of cards in the deck is 52 and the number of black cards is 26. So, the probability \(P(B)\) of drawing a black card in one draw is: \[P(B) = \frac{number \, of \, favorable \, outcomes}{total \, number \, of \, outcomes} = \frac{26}{52} = 0.5\]
02

Determine the probability of drawing a black card in the second draw

Since the card is replaced after the first draw and the deck is shuffled, the total number of cards remains the same, as well as the number of black cards. Therefore, the probability \(P(B')\) of drawing a black card in the second draw is also 0.5.
03

Calculate the total probability of drawing a black card twice with replacement

Since the two draws are independent events, the overall probability \(P(B and B')\) is the product of the probabilities of each event: \[P(B and B') = P(B) \times P(B') = 0.5 \times 0.5 = 0.25\]

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