Question

A fair coin is continually flipped until heads appears for the 10th time. Let X denote the number of tails that occur. Compute the probability mass function of X.

Answer 1

The probability mass function is expressed as:

P(x) = [(x+r-1)C(r-1)]*[p^r]*[(1-p)^x]

Explanation:

This is not a binomial distribution. It is actually a negative binomial distribution. The probability mass function is expressed below:

P(x) = [(x+r-1)C(r-1)]*[p^r]*[(1-p)^x]

where:

x = number of failures

r-1 = number of successes (10 in this scenario)

p = probability of a success

nCr = n!/[r!(n-r)!]

The main formula difference in the positive binomial versus negative binomial is this: With respect to the negative binomial, it is obviously known that the last event will be: when we reach our 10th "head", we stop .

Thus, the last flip will ALWAYS be a "head".