# Combinatorics Statistics Question

The problem I am working on is:

An academic department with five faculty members—Anderson, Box, Cox, Cramer, and Fisher—must select two of its members to serve on a personnel review committee. Because the work will be time-consuming, no one is anx-ious to serve, so it is decided that the representative will be selected by putting the names on identical pieces of paper and then randomly selecting two.

a.What is the probability that both Anderson and Box will be selected? [Hint:List the equally likely outcomes.]

b.What is the probability that at least one of the two members whose name begins with C is selected?

c. If the five faculty members have taught for 3, 6, 7, 10, and 14 years, respectively, at the university, what is the probability that the two chosen representatives have a total of at least 15 years’ teaching experience there?

For a), I figured that since probability of Anderson being chosen is $1/5$ and Box being chosen is $1/5$ the answer would simply be $2/5$. It isn't, though. It is $0.1$ How did they get that answer? I might need help with parts b) and c) as well.

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2/5 is the probability that either Anderson or Box would be selected as one person out of five. – Joe Z. Feb 1 '13 at 18:05
The problem asked you to make a list. So make a list. – André Nicolas Feb 1 '13 at 18:37

(a) You will have to consider the 10 different outcomes. Each combination of $"A-B", "B-Cox", "B-Cramer", ...$ is equally likely to be drawn. There are
$${5 \choose 2} = 10$$

10 ways to choose 2 people from a group of 5. Since we are only considering 1 possibility, the answer will be $\frac 1 {10} = 0.1$.

For (b), there are a few possibilities where at least 1 "C" member gets selected. Consider $$"Cox-A", "Cox-B", "Cox-Cramer", "Cox-F"$$ and $$"Cramer-A", "Cramer-B", "Cramer-Cox", "Cramer-F"$$ Notice that $"Cox-Cramer"$ appears twice, so one needs to be eliminated. Turns out there are 7 different possibilities out of 10...so we have $0.7$.

Alternatively, you can consider the different ways no "C" member gets selected. Turns out there are 3 ways $$"A-B", "A-F", "B-F"$$ resulting in $0.3$ chance of the compliment case happening. Taking $1 - 0.3$, you will still get $0.7$ as the answer.

For (c), I'd personally recommend taking the "compliment" of events. ie, how many combinations have LESS than 15 years? Turns out there are 4 combinations - $(3,6), (3,7), (3,10) , (6,7)$. You can use the answer from (a) and evaluate this to become $1 - \frac 4 {10} = 0.6.$

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(a) There are $\binom{5}{2}=10$ different ways of selecting 2 unique objects out of 5. Only 1 of them is success, hence the probability is $\frac{1}{10}$.

(b) It is 1-probability that no C's are selected, hence it is $1-\frac{3}{10}=\frac{7}{10}$

(c) same approach: 1-probability that the selected members have taught for 14 or less years, hence it is $1 - \frac{4}{10}=\frac{6}{10}$

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I think that it could be a possibility of what the persons population of their city or state to have any outcome to which you will need to find how to answer the question like for instants I live in Anderson and we could have a population of 2,364, so I can conclude that the possibility of someone in Anderson to think of the same number I'm thinking of will be 1/2364 so it could a numerous of numbers for the population so me and anyone can find out their probability of their city, state, and/or country.

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most simple way to understand this problem. (i just did this problem just now just now) lol

A, B, Co, Cr, and F exist. pick 1 of 5 at random. (1/5) pick another at random, but this time around there are only 4 choices. so (1/4) multiply the two values. (1/20)

BUT! that's considering that A is picked at first try then B on the second. you also have to consider the possibility that B would be picked first and etc.

so you add (1/20) + (1/20) = (2/20) = (1/10) = 0.1

pce

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This is a partial answer. Can you discuss how to generalize the approach so that the original poster can apply this principle to the other parts of the problem? (By the way, welcome to Math.SE!) – Cameron Buie Sep 21 '13 at 9:56