If in 6 trials, X is a binomial random variable which follows the relation 9P(x = 4) = P(x = 2) , then the probability of failure is...

Step-by-step solution

Let p be the probability of success and q be the probability of failure in a single trial, where p + q = 1 . For a binomial distribution with n = 6 trials, the probability of x successes is given by P(X = x) = ⁶C_ x p^ x q^ 6-x . Given the relation: 9P(X = 4) = P(X = 2) Substituting the binomial probabilities: 9 ⁶C₄ p⁴ q² = ⁶C₂ p² q⁴ Since ⁶C₄ = ⁶C₂ = 15 , we can cancel it from both sides: 9 p⁴ q² = p² q⁴ Dividing both sides by p² q² (assuming p, q 0 ): 9 p² = q² Taking the square root on both sides (since probabil