If X B(n, p) , then the value of P(X = k) P(X = k - 1) is

Step-by-step solution

For a binomial distribution X B(n, p) , the probability of k successes is given by P(X = k) = ^ n C_ k p^k q^ n-k , where q = 1 - p . Substituting k and k-1 , we get: P(X = k) = ^ n C_ k p^k q^ n-k P(X = k - 1) = ^ n C_ k-1 p^ k-1 q^ n-k+1 Taking the ratio of the two probabilities: P(X = k) P(X = k - 1) = ^ n C_ k p^k q^ n-k ^ n C_ k-1 p^ k-1 q^ n-k+1 P(X = k) P(X = k - 1) = ^ n C_ k ^ n C_ k-1 p q Using the property of combinations ^ n C_ r ^ n C_ r-1 = n - r + 1 r : P(X = k) P(X = k - 1) = ( n - k + 1 k ) p q Ans