Step-by-step Solution:

Since \( X \) and \( Y \) are independent Bernoulli random variables taking values in \( \{0, 1\} \), their product \( Z = XY \) will also strictly take values in \( \{0, 1\} \). Thus, \( Z \) follows a Bernoulli distribution. \( Z \) evaluates to 1 if and only if both \( X = 1 \) and \( Y = 1 \). Since \( X \) and \( Y \) are independent, we multiply their respective probabilities: \[ P(Z = 1) = P(X = 1 \cap Y = 1) = P(X = 1) \times P(Y = 1) \] \[ P(Z = 1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \] The probability that \( Z = 0 \) is simply the complement: \[ P(Z = 0) = 1 - P(Z = 1) = 1 - \frac{1}{4} = \frac{3}{4} \] Therefore, \( Z \) follows a Bernoulli distribution with \( P(Z=1)=1/4 \) and \( P(Z=0)=3/4 \).