Let S = w₁, w₂, w₃, w₄, w₅, w₆ be the sample space of a random experiment such that P(w_n) = 2^ n-1 63 for n = 1, 2, , 6 . The number of events A S such that P(A) 16 21 is _____.

Step-by-step solution

The probability of an event A is the sum of the probabilities of its elements. Given P(A) 16 21 = 48 63 . Let A^c be the complement of event A . Then P(A^c) = 1 - P(A) 1 - 48 63 = 15 63 . The probabilities of the elements are 1 63 , 2 63 , 4 63 , 8 63 , 16 63 , 32 63 . The sum of the numerators for the elements in A^c must be 15 . The available numerators are 1, 2, 4, 8, 16, 32 . If A^c contains w₅ or w₆ , the sum of numerators will be at least 16 , which is not allowed. Thus, A^c can only contain elements from w₁,