An endurance contest is being held with two independent grou

An endurance contest is being held with two independent groups of 14 participants. Individual participants in the contest drop out before the end of the contest with probability 0.16 (independently of other participants). What is the probability that at least 13 participants complete the endurance contest in one of the two groups, but not in both groups?

What is the probability that at least 13 participants complete the endurance contest in one of the two groups, but not in both groups?

Solution

Solution:-The probalility that a given individual drop out is 0.16.

The probability that a given individual stays in is 1-0.16 = 0.84.

The probability that an entire group of 14 completes the contest is (0.84)¹⁴.

The probability that a group of 14 has only one drop out is 14 X (0.16)X(0.84)¹³.

It is given that, an endurance contest is held with two independent groups.

so, the two events are independent events, hence the probability that atleast 13 in the group contest will be the addition of the above two cases.

That is ,(0.84)¹⁴+ 14X(0.16)(0.84)¹³

(0.84)¹³ (0.84+14X0.16)=0.31929.

The probabilty that either one of the groups does so and the other grouop doesnot is,that is the probability that a given group doesnot complete the contest with atleast 13 participants is given by

1- 0.31929=0.6807.

There are 2 ways for one group to finish with atleast 13 and the other not to participate in the contest .

That is the overall probablity that one of these 2 cases occuring is 2X0.31929X0.6807=0.43469.

The probability that atleast 13 participants complete the endurance contest in one of the two groups, but not in both groups is 0.43469.