Question

A card is drawn, with replacement, from a regular deck of cards 16 times. Let random variable X represent number of clubs among those 16 cards selected (there are 13 clubs in every deck; there are 52 cards in a deck). Find the variance of X,

Answer 1

To determine the variance of the random variable \( X \), which represents the number of clubs drawn from a standard deck of cards in 16 draws with replacement, we can model this scenario using a binomial distribution.

### Parameters of the binomial distribution:
- The number of trials \( n \) is 16 (since we draw 16 cards).
- The probability of success \( p \) (drawing a club) is given by the fraction of clubs in the deck:
\[
p = \frac{13}{52} = \frac{1}{4}.
\]

### Variance of a binomial distribution:
The variance \( \text{Var}(X) \) of a binomially distributed random variable \( X \) can be calculated using the formula:
\[
\text{Var}(X) = n p (1 - p),
\]
where \( n \) is the number of trials and \( p \) is the probability of success.

### Applying the values:
Here, substituting \( n = 16 \) and \( p = \frac{1}{4} \):
\[
1 - p = 1 - \frac{1}{4} = \frac{3}{4}.
\]

Now, we can calculate the variance:
\[
\text{Var}(X) = 16 \cdot \frac{1}{4} \cdot \frac{3}{4}.
\]

Calculating:
\[
\text{Var}(X) = 16 \cdot \frac{1}{4} \cdot \frac{3}{4} = 16 \cdot \frac{3}{16} = 3.
\]

Therefore, the variance of \( X \) is:
\[
\boxed{3}.
\]