A biased six-sided die is such that the probability of getting any specific odd number is p , and the probability of getting any specific even number is q . The probability of getting a prime number on a single roll of this die is 5 12 . The die is rolled n times. Let X be the random variable representing the number of times an odd prime number is obtained. If P(X=1) = P(X=2) , then the variance of X is

Step-by-step solution

Let the probability of getting any specific odd number be p and any specific even number be q . Since there are 3 odd and 3 even numbers on a 6-sided die: 3p + 3q = 1 p + q = 1 3 The prime numbers on a die are 2, 3, and 5. Here, 2 is even, while 3 and 5 are odd. P( prime ) = P(2) + P(3) + P(5) = q + 2p = 5 12 Subtracting p + q = 4 12 from 2p + q = 5 12 gives: p = 1 12 The success event is getting an odd prime (3 or 5). Its probability is: P( odd prime ) = 2p = 2 12 = 1 6 Let X follow a binomial distribution with pa