Solving Probability Questions: Bayes' Formula & Variables

[Maxwell]

Hey guys, I'm taking a probability course and I'm having some trouble with 2 questions:

1) Suppose 10% of a company's life insurance policy holders are smokers. The rest are non-smokers. For each non-smoker, the probability of dying during the year is 1% compared to 5% for smokers. Given that a policy holder has died, what is the chance that the policy holder is a smoker?

Ok, now for this one I had a feeling I should use Bayes' formula. The problem I'm having is assigning variables. This is what I did:

A1 = {smoker}
A2 = {non-smoker}
B1 = {smoker dying}
B2 = {non-smoker dying}
C = 10%

Pr{A1} = 10/100 = .1
Pr{A2} = 90/100 = .9
Pr{B1} = 5%
Pr{B2} = 1%

I'm not sure if these are even set up right, let alone how to put them into Bayes' formula. Also, how do I write what I am looking for?

I know that a policy holder died -- The probability that this person was a smoker is 10%. This smoker had a 5% chance of dying during the year.

I'm really stuck, though...


The second problem:

Three missiles, whose probabilities of not hitting a target are 0.3, 0.2, and 0.1, respectively, are fired at a target. Assuming independence, what is the probability that the target is hit by all of the three missiles?

Now for this problem, I assigned a variable to each missile:

b1 = {missile 1}
b2 = {missile 2}
b3 = {missile 3}

Then,

a1 = {hit target}
a2 = {not hitting target}

So,

Pr{b1 | a1} = 0.7
Pr{b1 | a2} = 0.3

Pr{b2 | a1} = 0.8
Pr{b2 | a2} = 0.2

Pr{b3 | a1} = 0.9
Pr{b3 | a2} = 0.1

Now, I think I'm looking for something like Pr(b1 & b2 & b3 | a1). Am I right?

If so, 0.7 x 0.8 x 0.9 = .504

Is this correct?

Thank you for your help.

 

[Office_Shredder]

I'm not sure if these are even set up right, let alone how to put them into Bayes' formula. Also, how do I write what I am looking for?

I know that a policy holder died -- The probability that this person was a smoker is 10%. This smoker had a 5% chance of dying during the year.

I'm really stuck, though...

If the probability of this person being a smoker is 10%, you just answered the question (hint: that's not the answer).

Try calculating the probability of a smoker dying, and the probability of a non-smoker dying (if you're having trouble, just assume it's a group of ten people).

Then compare the odds of it being a smoker vs. a non-smoker

If so, 0.7 x 0.8 x 0.9 = .504

Is this correct?

Thank you for your help.

Yes, this one is correct

 

[Maxwell]

Thanks Office_Shredder.

So I'm trying to find the probability of a smoker dying. Pr{B1} = 5% isn't that probability?

I think the probability that a smoker died is: Pr{B1 | A1}

And,

Probability that a non-smoker died: Pr{B2 | A2}

Is this the right way of mathematically stating what I am asking? I think the real problem I'm having with all of the Bayes' formula questions is setting up my variables correctly and then finding the corresponding probabilities.