A law firm has thirteen senior and nine junior partners. A committee of 6 partners is selected at random to represent the firm at a conference. What is the probability that at least one of the junior partners is on the committee?
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I will do this last one, but then you must show us that you have some kind of method for doing these.
number of possible committees
= C(22,6) = 74613
number with NO junior partners = C(13,6) = 1716
so the number with at least one junior = 74613-1716 = 72897
prob of at least one junior = 72897/74613 = 6627/6783= 2209/2261 = appr .977
i want help understanding how to solve the problems, not just the answers. I understand how to get the denominator, by getting the total and nCr the sample, but i cannot figure out the numerator when it comes to solving these probability problems.
To find the probability that at least one of the junior partners is on the committee, we will first calculate the probability that none of the junior partners are selected, and then subtract that probability from 1.
The total number of partners in the law firm is 13 seniors + 9 juniors = 22 partners.
The total number of possible committees of size 6 that can be selected from the 22 partners is given by the combination formula:
C(22, 6) = 22! / (6!(22-6)!) = 74613
Now let's calculate the number of committees that can be selected without any junior partners:
There are 13 seniors, so we need to select 6 members from the 13 seniors, which is given by the combination formula:
C(13, 6) = 13! / (6!(13-6)!) = 1716
Therefore, the number of committees without any junior partners is 1716.
Now, we can calculate the probability that none of the junior partners are selected:
P(no junior partner) = (number of committees without any junior partners) / (total number of possible committees)
P(no junior partner) = 1716 / 74613 ≈ 0.023
Finally, we can find the probability that at least one of the junior partners is on the committee:
P(at least one junior partner) = 1 - P(no junior partner)
P(at least one junior partner) = 1 - 0.023 ≈ 0.977
Therefore, the probability that at least one of the junior partners is on the committee is approximately 0.977 or 97.7%.