I am in need of some explanation as for whatever reason I just can't wrap my head around a problem.
The question basically breaks down like this:
- There are $8$ people on a jury ($3$ men and $5$ women).
- Each person can vote either yes or no
The question basically asks "what is the probability that the vote would split along gender lines (The $5$ women voting Yes, the $3$ men voting no)?"
The answer in the back of the textbook is: $\frac{1}{56}$.
The $1$ comes because there is only way of having all the women voting yes and all the men voting no. That is pretty obvious and makes sense to me.
The trouble I am having is trying to understand the $56$. So in probability the bottom number is the number of all possible outcomes. In this case I believe that would be: all the possible ways $8$ could people vote yes or no".
My calculation initially for all possible outcomes was: $2^8 = 256$. My reasoning was there are $8$ people and each person can vote either yes or no, so each person has two options. Therefore, Person$1$ has $2$ options, Person$2$ has $2$ options, Person$3$ has $2$ options. Which is $2×2×2×2×2×2×2×2$ ($2^8$).
So my final answer would be: $\frac{1}{256}$. Clearly I'm missing something since the textbooks answer is $\frac{1}{56}$.
I can see that maybe that $56$ comes through Permutations. That is to say $P(8,2) = 56$. The way I'd interrupt that is "how many ways can $8$ people be put in two groups (i.e the "yes" group and the "no" group). I'm not entirely sure if that's how the textbook solved the problem.
Would love if someone could possibly help clarify why $2^8$ isn't the correct answer. As well as to help clarify why $56$ is the right answer.
Edit: Word for word the full question:
The following case occurred in
Gainesville, Florida. The eight-member Human Relations
Advisory Board considered the complaint of a
woman who claimed discrimination, based on her
gender, on the part of a local surveying company. The
board, composed of five women and three men, voted
5–3 in favor of the plaintiff, the five women voting
for the plaintiff and the three men against. The attorney
representing the company appealed the board’s
decision by claiming gender bias on the part of the
board members. If the vote in favor of the plaintiff
was 5–3 and the board members were not biased by
gender, what is the probability that the vote would
split along gender lines (five women for, three men
against)?
Best Answer
If you assume (as you are usually expected to in these problems, even though it is silly) that each person votes independently of the others, and votes yes with probability $1/2$, there are only $2$ of the $256$ voting results that are split by gender, so the chance is $2/2^8=1/128$ The factor two compared to your result is because I interpreted the parenthetical remark to also allow all the women to vote No and all the men to vote Yes. This is an English question, not a math question. Your math is fine and I think the English is debatable. As ${8 \choose 3} = 56$ I suspect what the question is really asking is "Given the vote is $5-3$ Yes, what is the chance that all the women voted yes and all the men voted No" If so, either the book is badly written or you did not post the correct problem.