Question:
A card is drawn from a pack of cards successively 5 times by replacing the card drawn in the pack of cards. What is the mean of the number of red cards drawn?
A card is drawn from a pack of cards successively 5 times by replacing the card drawn in the pack of cards. What is the mean of the number of red cards drawn?
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When events are repeated independently with the same probability, use the binomial distribution. The mean is simply \( n \cdot p \).
Updated On: May 27, 2025
- 3
- 2.5
- 3.5
- 2
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The Correct Option is B
Solution and Explanation
Step 1: Understand the experiment.
Each draw is independent because the card is replaced. The probability of drawing a red card (hearts or diamonds) from a standard 52-card deck is: \[ P(\text{red}) = \frac{26}{52} = \frac{1}{2} \]
Step 2: Use the binomial distribution.
Let \( X \) be the number of red cards drawn in 5 trials. This is a binomial distribution: \[ X \sim \text{Binomial}(n = 5, p = 0.5) \]
Step 3: Calculate the mean.
Mean of a binomial distribution is: \[ \mu = n \cdot p = 5 \cdot \frac{1}{2} = 2.5 \]
Each draw is independent because the card is replaced. The probability of drawing a red card (hearts or diamonds) from a standard 52-card deck is: \[ P(\text{red}) = \frac{26}{52} = \frac{1}{2} \]
Step 2: Use the binomial distribution.
Let \( X \) be the number of red cards drawn in 5 trials. This is a binomial distribution: \[ X \sim \text{Binomial}(n = 5, p = 0.5) \]
Step 3: Calculate the mean.
Mean of a binomial distribution is: \[ \mu = n \cdot p = 5 \cdot \frac{1}{2} = 2.5 \]
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