Conditional Probability Questions

1. Sep 18, 2012

daneault23

1. You give a friend a letter to mail. He forgets to mail it with probability 0.2. Given that
he mails it, the Post Office delivers it with probability 0.9. Given that the letter was not
delivered, what’s the probability that it was not mailed?

2. I assume I'm supposed to use Bayes Formula, but I'm confused as to what we really know in order to solve the problem.

3. I used A= not mailed, B= not delivered. I have this equation, P(A given B) = (P(B given A) *P(A))/ P(B given A)*P(A) + P(B given A complement)*P(A complement)
which equals (?*.20)/?*.20 + ? * .80

2. Sep 18, 2012

jbunniii

P(B given A) = P(not delivered given not mailed). This part should be a no-brainer.

3. Sep 18, 2012

daneault23

Okay well if that is 0 then the answer to the whole problem is 0, and that doesn't seem to be right. By the way the question was asked, it would then say that he never forgets to mail it, which isn't right given the problem. I think I set it up wrong somehow.

4. Sep 18, 2012

jbunniii

No, it's not zero. Remember there are "nots" on both events:

P(NOT delivered, given NOT mailed)

Can something be delivered if it is not mailed?

5. Sep 18, 2012

daneault23

I don't get what you're saying. If something isn't mailed, there's no way it can be delivered.. theres nothing to deliver..

6. Sep 18, 2012

jbunniii

Right, so what is the probability that it is not delivered, given that it was not mailed?

7. Sep 18, 2012

daneault23

1

So it would look this correct?

(1)(.20)
----------------------- = 5/7
(1)(.20) + (.1)(.80)

8. Sep 18, 2012

jbunniii

Yes, that looks right to me.

9. Sep 18, 2012

Ray Vickson

Start with proper notation, so you an keep things straight; for example, you can immediately see what is being spoken of if you let M = {mailed}, D ={delivered}, and their complements NM = {not mailed}, ND = {not delivered}. You are given P(NM) = 0.2 and P(D|M) = 0.9, and you want to compute P(NM|ND).

Your answer will be P(NM and ND)/P(ND). How can you compute the numerator? How can you compute the denominator?

Sometimes, people find it easier to do such problems by direct counting, avoiding Bayes formulas until they have a better understanding of them (which seems to be your situation). So, here is how some people would do it. Imagine repeating this experiment 1,000,000 times. In 200,000 of these experiments the letter is not mailed, so in 800,000 of them it IS mailed. In all 200,000 not-mailed experiments the letter is not, of course, delivered. In the 800,000 'mailed' experiments what percentage of them get delivered? What number of them get delivered? So, out of the original 1,000,000 letters, how many letters are delivered and how many are not delivered? Among those that were not delivered, how many were mailed? Those two figures allow you to compute P(NM|ND) directly.

RGV