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: \[ \boxed{3}. \]