A town council considers the question of closing down an "adult" theater. The five men on the council all vote against this and the three women vote in favor. What is the probability that we would get this result
(a) if the council members determined their votes by flipping a coin?
(b) if we assigned the five "no" votes to council members chosen at random?
I'm guessing for part (a), I should multiply (1/2)^5 by (1/2)^3, but is there something else that I'm missing or am I completely missing the point? I'm not too sure how to start on part (b), though. Any help would be appreciated.
Thanks :)
(a) if the council members determined their votes by flipping a coin?
(b) if we assigned the five "no" votes to council members chosen at random?
I'm guessing for part (a), I should multiply (1/2)^5 by (1/2)^3, but is there something else that I'm missing or am I completely missing the point? I'm not too sure how to start on part (b), though. Any help would be appreciated.
Thanks :)
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For part (a) you are right if "this result" means exactly that the 5 men and 3 women
voted the way they did, since each vote is 1/2 - 1/2 either way when using a coin flip.
If it means "any 5-3 result", see below.
For (b):
There are 8c5 = 8c3 = 8 * 7 * 6 / 3! = 56 ways to pick
the 5 no (or alternatively 3 yes) votes.
[ If you choose 3 yes votes, the 8 - 3 = 5 must be no votes automatically,
and vice versa, so it's easier to figure the smaller group. ]
8 choices for first yes vote
then 7 for the second, since no one votes twice,
and 6 for the third.
Divide by 3! since the same 3 in a different order is considered the same
and there are 3 * 2 * 1 = 3! different orders.
So Pr(it came out exactly this way) = 1 / 56 (given that it was 5-3)
For coin flip voting:
If all possible voting results are considered equally likely,
then there 256 or them (2^8), and Pr (5 - 3 split) = 56 / 256 or about 22%
8-0 or 0-8 is 1/256 (each)
7-1 or 1-7 is 8/256 = 1/64 (each)
2-6 or 6-2 is 28/256 = about 11% (each)
3-5 or 5-3 is 56/256 = about 22% (each)
4-4 is 70/256 = about 27%
It's not the math which is confusing here - it's the meaning of the words!
voted the way they did, since each vote is 1/2 - 1/2 either way when using a coin flip.
If it means "any 5-3 result", see below.
For (b):
There are 8c5 = 8c3 = 8 * 7 * 6 / 3! = 56 ways to pick
the 5 no (or alternatively 3 yes) votes.
[ If you choose 3 yes votes, the 8 - 3 = 5 must be no votes automatically,
and vice versa, so it's easier to figure the smaller group. ]
8 choices for first yes vote
then 7 for the second, since no one votes twice,
and 6 for the third.
Divide by 3! since the same 3 in a different order is considered the same
and there are 3 * 2 * 1 = 3! different orders.
So Pr(it came out exactly this way) = 1 / 56 (given that it was 5-3)
For coin flip voting:
If all possible voting results are considered equally likely,
then there 256 or them (2^8), and Pr (5 - 3 split) = 56 / 256 or about 22%
8-0 or 0-8 is 1/256 (each)
7-1 or 1-7 is 8/256 = 1/64 (each)
2-6 or 6-2 is 28/256 = about 11% (each)
3-5 or 5-3 is 56/256 = about 22% (each)
4-4 is 70/256 = about 27%
It's not the math which is confusing here - it's the meaning of the words!