# Probability of a group of people voting yes or no

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)?

• I agree. Gender doesn't matter. What are odds that jurors 1-3 vote yes, and 4-8 vote no? 1/256 – JoeTaxpayer Jun 30 '15 at 3:13
• Maybe it's just a typo error------- missing the 2 from 256 – Peter Jun 30 '15 at 3:37

## 3 Answers

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.

• Thanks for the reply! I'll just go ahead and post the exact question in an edit (it's too long to post as a reply). Maybe that will reveal something. – Shawn Jun 30 '15 at 3:47
• Your post of the exact question supports my supposition. You are given the $5-3$ Yes result and are asking what is the chance (assuming independence) that it was all the women who voted Yes. Given that the vote was $5-3$ there are only $56$ possible outcomes, not $256$. That is the heart of your problem. – Ross Millikan Jun 30 '15 at 3:54
• Ah I understand the flaw in my thinking. Thanks very much @RossMillikan that makes a lot of sense! So just to be clear if the vote is 5yes-3no, I'm really looking for how many different ways I can obtain the outcome 5yes-3no which would be: (8 choose 5)*(3 choose 3) = 56. So the probability would be 1/56. In my original line of thinking I was worried about all possible vote out comes, which was my problem. – Shawn Jun 30 '15 at 4:03
• That is correct. You were given that the result was $5-3$ Yes and your calculation did not incorporate that information. – Ross Millikan Jun 30 '15 at 4:09

Because in the question we only care about gender. We don't treat "John votes Yes and the rest vote No" and "Smith votes Yes and the rest vote No" as two different cases. That is the case "a man votes Yes, 2 men vote No, all the women vote No".

So I think I figured out the answer and how to solve it. Taking it from either side you know 1 case is 5 said yes and 3 said no. Assuming all the women said yes that means the men said no. You start with the nos because the math is easier. The chance the first guy said yno is 3/8 then the chance the second guy said no is 2/7 and the chance the third guy said no is 1/6. So now you have 3/8 * 2/7 * 1/6. Once you reduce it all down you get 1/56th is the probability all the men said no. To validate it would be the same probability that all the women say yes is similar (5/8 * 4/7 * 3/6 * 2/5 * 1/4) which reducing down and multiplying gives you 1/56th