Probability: P(A|B') for Married Couples

[ViolentCorpse]

Homework Statement

For married couples, the probability that the husband will vote on a bond referendum is 0.21, the probability that his wife will vote in the referendum is 0.28 and the probability that both the husband and wife will vote is 0.15. What is the probability that a husband will vote, given that his wife does not vote?

Homework Equations

If A is probability that husband votes, and B the probability that the wife votes, the probability that husband votes given that the wife does NOT should be:

P(A|B')=P(A n B')/P(B')

The Attempt at a Solution

But, I have absolutely no idea how I can find (A n B') from the given data...

I'll be very grateful for your help.

 

[D H]

Hint: You know both $P(A \cap B)$ and $P(A)$.

 

[PeroK]

To understand the problem better, you could write down all the probabilities of the husband and wife voting or not:

P(both vote ) = 0.15
P(Wife votes, husband doesn't) = ?

Etc.

 

[Ray Vickson]

Homework Statement

For married couples, the probability that the husband will vote on a bond referendum is 0.21, the probability that his wife will vote in the referendum is 0.28 and the probability that both the husband and wife will vote is 0.15. What is the probability that a husband will vote, given that his wife does not vote?

Homework Equations

If A is probability that husband votes, and B the probability that the wife votes, the probability that husband votes given that the wife does NOT should be:

P(A|B')=P(A n B')/P(B')

The Attempt at a Solution

But, I have absolutely no idea how I can find (A n B') from the given data...

I'll be very grateful for your help.

Step 1: draw a Venn diagram. What do the various sub-regions represent? How would you find their probabilities?

 

[D H]

To understand the problem better, you could write down all the probabilities of the husband and wife voting or not:

P(both vote ) = 0.15
P(Wife votes, husband doesn't) = ?

Etc.

A convenient way to do this is with a contingency table (aka a confusion matrix).

            |    Husband       | Wife
            |  Vote    ~Vote   | Total
-------------------------------+------
Wife   Vote | P(A∩B)  P(~A∩B)  | P(B)
      ~Vote | P(A∩~B) P(~A∩~B) | P(~B)
-------------------------------+------
Husb. Total |  P(A)    P(~A)   |  1.0

You have the information at hand to completely populate this table.

 

[ViolentCorpse]

I know P(A), P(B) and P(A n B), but I don't have a clue how I can use any of these probabilities to find P(A n B').

Sorry guys. :/

 

[D H]

The event "husband and wife voted" and "husband voted, wife did not vote" are mutually exclusive events. Are there any other events that collectively form the event "husband voted"? How would you express this mathematically?

 

[ViolentCorpse]

Alright so I drew Venn diagrams and thought hard and I think that P(A n B') is actually P(A)-P(A n B) and also P(A)=P[(AnB')U(AnB)].

I've got the correct answer using this relation, but since I haven't found any equations like that in my book yet, I'm not confident that these equations are valid.

 

[D H]

That is exactly correct. Those are the relations you need to answer the problem.

 

[ViolentCorpse]

Wonderful. Thank you so much, DH and everyone else! :)

 

[HallsofIvy]

Another way of looking at it: imagine 100 couples. Then 21 husbands vote and 100- 21= 79 do not. Similarly 28 wives vote for it and 100- 28= 72 do not. There are 15 couples, 15 husbands and 15 wives, included in those. So there are 21- 15= 6 husbands whose wives do NOT vote and 28- 15= 13 wives whose husbands do not vote. So the probability a wife votes when her husband does not is 13/100= 0.13

 

[ViolentCorpse]

Another way of looking at it: imagine 100 couples. Then 21 husbands vote and 100- 21= 79 do not. Similarly 28 wives vote for it and 100- 28= 72 do not. There are 15 couples, 15 husbands and 15 wives, included in those. So there are 21- 15= 6 husbands whose wives do NOT vote and 28- 15= 13 wives whose husbands do not vote. So the probability a wife votes when her husband does not is 13/100= 0.13

I'm sorry but I lost you at that part..

Thanks for further trying to clarify it, HallsofIvy. :)