Quote:
Example #1
Q: There are 8 employees including Bob and Rachel. If 2 employees are to be randomly chosen to form a committee, what is the probability that the committee includes both Bob and Rachel?
Solution:
1) combinatorial approach: The total number of possible committees is N=C82N=C28. The number of possible committee that includes both Bob and Rachel is n=1n=1.
P=nN=1C82=128P=nN=1C28=128
2) reversal combinatorial approach: Instead of counting probability of occurrence of certain event, sometimes it is better to calculate the probability of the opposite and then use formula p = 1 - q. The total number of possible committees is N=C82N=C28. The number of possible committee that does not includes both Bob and Rachel is:
m=C62+2∗C61m=C26+2∗C16 where,
C62C26 - the number of committees formed from 6 other people.
2∗C612∗C16 - the number of committees formed from Rob or Rachel and one out of 6 other people.
P=1−mN=1−C62+2∗C61C82P=1−mN=1−C26+2∗C16C28
P=1−15+2∗628=1−2728=128
Hello Walker, thanks for the great post!
I don't understand the second part of (2) and was hoping you could help answer a specific question.
For the number of committees formed without one of Bob or Rachel, I'm confused about the value of n in nCr. You have used n = 6 (6C1*2). Shouldn't we instead use 7C1*2, where 7 represents the total number of people without Bob or Rachel and we pick out 2?