Question

A die with numbers 1 to 6 is biased such that \( P(2) = \frac{3}{10} \) and the probability of other numbers is equal. Find the mean of the number of times number 2 appears on the die, if the die is thrown twice.

Hint:

For a binomial distribution \( B(n, p) \), the mean is given by \( E(X) = n p \).

Answer 1

Step 1: Define the random variable. Let \( X \) be the number of times the number 2 appears in two throws of the die. Since each throw is independent, \( X \) follows a binomial distribution: \[ X \sim B(n, p), \] where: - \( n = 2 \) (number of trials), - \( p = P(2) = \frac{3}{10} \) (probability of success in each trial). 

Step 2: Compute the expected value (mean). The expectation for a binomially distributed random variable is given by: \[ E(X) = n p. \] Substituting values: \[ E(X) = 2 \times \frac{3}{10} = \frac{6}{10} = 0.6. \] 

Final Answer: \[ \text{Mean number of times 2 appears} = 0.6. \]