Question

an endurance contest is being held with two independent groups of 8 participants. individual participants in the contest drop out before the end of the contest with probability 0.3 (independently of other participants). what is the probability that at least 7 participants complete the endurance contest in one of the two groups, but not in both groups?
choose the correct answer below.
a. 0.3802
b. 0.6289
c. 0.7605
d. 0.1901

Explanation:

Step1: Calculate the probability that at least 7 participants complete in a group

The probability that a participant completes the contest is \(p = 1 - 0.3=0.7\).
Using the binomial probability formula \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 8\), \(C(n,k)=\frac{n!}{k!(n - k)!}\)
For \(k = 7\): \(P(X = 7)=C(8,7)\times0.7^{7}\times0.3^{1}=\frac{8!}{7!(8 - 7)!}\times0.7^{7}\times0.3=8\times0.7^{7}\times0.3\)
For \(k = 8\): \(P(X = 8)=C(8,8)\times0.7^{8}\times0.3^{0}=1\times0.7^{8}\times1 = 0.7^{8}\)
The probability that at least 7 participants complete in a group is \(P=P(X = 7)+P(X = 8)=8\times0.7^{7}\times0.3+0.7^{8}=0.7^{7}(8\times0.3 + 0.7)=0.7^{7}(2.4 + 0.7)=0.7^{7}\times3.1\approx0.2553\)

Step2: Calculate the required probability

Let \(A\) be the event that at least 7 participants complete in the first - group and not in the second - group, and \(B\) be the event that at least 7 participants complete in the second - group and not in the first - group.
Since the two groups are independent, \(P(A)=P\times(1 - P)\) and \(P(B)=(1 - P)\times P\)
The probability that at least 7 participants complete in one of the two groups, but not in both groups is \(P(A + B)=2\times P\times(1 - P)\)
Substitute \(P\approx0.2553\) into the formula: \(2\times0.2553\times(1 - 0.2553)=2\times0.2553\times0.7447\approx0.3802\)

Answer:

A. 0.3802