An 8-sided biased die has faces numbered 1 to 8. The probability of getting any specific face depends only on its remainder when divided by 3. Let p_i be the probability of rolling face i . It is given that 3 p_i for i 3, 6 equals 2 p_j for j 1, 4, 7 equals 6 p_l for l 2, 5, 8 . If the die is thrown 8 times, let X be the random variable representing the number of times a prime number appears. The mean of X is

Step-by-step solution

Let p₃ = p₆ = a , p₁ = p₄ = p₇ = b , and p₂ = p₅ = p₈ = c . Given that 3a = 2b = 6c = k , we have a = k 3 , b = k 2 , and c = k 6 . The sum of all probabilities must be 1, so: 2a + 3b + 3c = 1 2 ( k 3 ) + 3 ( k 2 ) + 3 ( k 6 ) = 1 2k 3 + 3k 2 + k 2 = 1 2k 3 + 2k = 1 8k 3 = 1 k = 3 8 Thus, a = 1 8 , b = 3 16 , and c = 1 16 . The prime numbers on an 8-sided die are 2, 3, 5, and 7. The probability of getting a prime number is: P( prime ) = p₂ + p₃ + p₅ + p₇ = c + a + c + b = a + b + 2c P( prime ) = 1 8 + 3 16 + 2 ( 1