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Chapter 11: Problem 19

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 red card each time.

Short Answer

Expert verified
The probability of drawing a red card twice in a row, when the card is replaced each time, is \( \frac{1}{4} \) or 0.25.

Step by step solution

01

Identify the total number of outcomes and successful outcomes

A standard deck has 52 cards, with 26 of them being red (13 Hearts and 13 Diamonds). Thus, the total number of outcomes is 52, and the successful outcomes (drawing a red card) is 26. The probability of one event is calculated by dividing the number of successful outcomes by the total number of outcomes. In this case, the probability of drawing a red card once, \( P(Red) \), is \( \frac{26}{52} = \frac{1}{2} \).
02

Apply the definition of independent events

As the card was replaced, the two card draws are independent events. The probability of two independent events both happening is the product of the probabilities of each event happening individually. This gives the formula \( P(Red and Red) = P(Red) \times P(Red) \).
03

Calculate the final probability

Substituting the previously calculated probability into the formula, \( P(Red and Red) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). So, the probability of drawing a red card twice in a row when the card is replaced each time is \( \frac{1}{4} \).

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