A stack of playing cards contains 4 jacks, 5 queens, 3 kings, and 3 aces. Two cards will be randomly selected from the stack. What is the probability that a queen is chosen and replaced, and then a queen is chosen again?
$\frac{1}{9}$
$\frac{2}{21}$
$\frac{16}{225}$
$\frac{4}{9}$
Answer (2)
Calculate the total number of cards: 4 + 5 + 3 + 3 = 15 .
Find the probability of drawing a queen on the first draw: 15 5 = 3 1 .
Find the probability of drawing a queen on the second draw (after replacement): 15 5 = 3 1 .
Multiply the probabilities: 3 1 × 3 1 = 9 1 .
Explanation
Understand the problem We are given a stack of playing cards with 4 jacks, 5 queens, 3 kings, and 3 aces. We want to find the probability of drawing a queen, replacing it, and then drawing a queen again.
Calculate the total number of cards First, let's calculate the total number of cards in the stack. We have 4 jacks + 5 queens + 3 kings + 3 aces = 15 cards in total.
Calculate the probability of drawing a queen on the first draw The probability of drawing a queen on the first draw is the number of queens divided by the total number of cards, which is 15 5 = 3 1 . Since we replace the card, the composition of the deck remains the same for the second draw.
Calculate the probability of drawing a queen on the second draw The probability of drawing a queen on the second draw is also 15 5 = 3 1 , because we replaced the first card.
Calculate the probability of drawing a queen on both draws To find the probability of both events happening, we multiply the probabilities of each event: P ( Queen then Queen ) = P ( Queen on 1st draw ) × P ( Queen on 2nd draw ) = 3 1 × 3 1 = 9 1 .
State the final answer Therefore, the probability that a queen is chosen and replaced, and then a queen is chosen again is 9 1 .
Examples
This type of probability calculation is useful in scenarios like card games or any situation where you're drawing items from a set with replacement. For example, if you're running a raffle where you draw a ticket, announce the winner, and then put the ticket back in before drawing again, this calculation helps you determine the likelihood of selecting the same person twice in a row.
The probability of drawing a queen, replacing it, and then drawing a queen again from the stack of playing cards is 9 1 . This is calculated by finding the probability of drawing a queen each time, which is 3 1 for both draws, and multiplying the probabilities together. The correct answer is 9 1 .
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