Conditional Probability Problem

According to a survey, a few years ago, 51% of the American respondents believed the Social Security system would be secure in 20 years. Of the respondents who were 40 or older, 60% believed the system would be secure in 20 years. Of the people surveyed, 52% were under age 40.

a) If one respondent is selected randomly, and the person selected believes the Social Security system will be secure in 20 years, what is the probability that the person is 40 years old or older?

answer is 56.47%

b) What is the probability that the person is younger than 40 and believes that the Social Security system will be secure in 20 years?

answer is 22.20%

I've tried so hard and I have no idea how they got these answers

Re: Conditional Probability Problem

Hey ttgtl.

The first step for these kind of problems is to translate the information to probabilities. Given what you have stated, you have P(Social Security in 20) = 0.51 and P(Age > 40 AND Secure in 20) = 0.6 with P(Age < 40) = 0.52

Hint: Use these along with the conditional probability formula P(A|B) = P(A and B)/P(B) along with P(not A) = 1 - P(A).

Re: Conditional Probability Problem

I used those formulas, they don't give me the correct answers

either that or I really do not understand how the formula works with independent probabilities

Re: Conditional Probability Problem

Show us all of your calculations so we can see what you did and what you were thinking please.

Re: Conditional Probability Problem

a) P(A|B) = 0.60 * 0.51 / 0.51

b) P(A|B) = 0.52 * 0.51 / 0.51

Re: Conditional Probability Problem

can someone please tell me what I need to do

Re: Conditional Probability Problem

So for these problems, we have to translate the words into symbolic probabilities. In a) I interpret this as to asking for P(Age > 40|Social Security OK in 20 years). This means that if A = Age > 40 and B = SS OK in 20 years then

P(A|B) = P(A and B)/P(B)

Also I made a mistake in my 2nd post: it should be P(Secure in 20|Age > 40) = 0.6.

So we are trying to find P(Age > 40|SS OK in 20) = P(Age > 40 AND SS OK in 20)/P(SS OK in 20) = 0.6*0.48/0.51 = 0.5647059 which is what the answer is from your book.

Basically I used the property P(A|B) = P(B|A)*P(A)/P(B) = P(A and B)/P(A) * P(A)/P(B) = P(A and B)/P(B) = P(A|B).

It's a little tricky but it works.

See if you can do the second one and I'll provide some hints if you need them.

Re: Conditional Probability Problem

I used 0.51 - 0.6*0.48 = 0.222

Re: Conditional Probability Problem

For b) you have the probability P(Age < 40 and Security OK in 20) but you have the condition where

P(Age >= 40 and Security OK) + P(Age < 40 and Security OK) = P(Security OK) = 0.51 which implies

P(Age < 40 and Security OK) = 0.51 - P(Age >= 40 and Security OK) = 0.51 - 0.6*0.48 = 0.222 which is what you got.

Once you get the hang of the probability axioms, it will become a lot easier: just remember to use the axioms as your intuition and not your gut or mental intuition and you will be OK.