A fair n faced die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is n 9 , then n = (where n N )

Step-by-step solution

The die has n faces with outcomes 1, 2, , n . A success occurs when rolling a number less than n , of which there are n - 1 favorable outcomes. The probability of success in a single toss is p = n-1 n . Let X denote the number of tosses until a success. Since X follows a geometric distribution, its expected value is E[X] = 1 p = n n-1 . Given that E[X] = n 9 , we set n n-1 = n 9 . For n 2 , dividing both sides by n yields 1 n-1 = 1 9 , so n - 1 = 9 and n = 10 . The value of n is 10 , corresponding to answer choice