Problem 205SE

An experiment consists of tossing a fair die until a 6 occurs four times. What is the probability that the process ends after exactly ten tosses with a 6 occurring on the ninth and tenth tosses?

Answer:

Step 1 of 1:

An experiment consists of tossing a fair die until a six occurs 4 times.

We need to find the probability that the process ends after exactly 10 tosses with a six occurring on the 9th and 10th tosses.

Sample space of fair die

Hence, from the sample space the probability of occurrence of six,

According to the question, out of 4 times in 10 tosses, six occurs exactly in ninth and tenth tosses and other two six can occur in any of first 8 tosses.

Hence first two six occurs in any of first 8 tosses is random and hence follows the binomial distribution with parameter

Let represent the number of tosses till first two sixes occur.

A random variable is said to have a binomial probability distribution based on trials with success probability if and only if

………….(1)

……..(1)

The probability of getting sixes on the 9th and 10th tosses,

………(2)

Therefore the required probability using equation (1) and (2),

Hence the probability that the process ends after exactly 10 tosses with a six occurring on the 9th and 10th tosses is