Conditional Probability Question Help

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juniet0331

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I have been trying to figure this question out and I may be overthinking it. I could really use some help and guidance.

Suppose that 24% of people suffer an acute breathing illness such as COPD by age 65. Research shows that 60% of smokers and 15% of non-smokers suffer from an acute breathing illness by age 65. In a certain office, 20% of the staff smokes cigarettes.

Let S = person smokes

B = person suffers acute breathing illness by age 65

1. Find the probability that a person in this office both smokes and will suffer an acute breathing illness by age 65.

I worked it out first that
P(B) =.24, P(S given B) = .60 and P(S) =.20

Issues:
1) I am not entirely sure if I classified the 60% correctly. I am having a hard time distinguishing if this would be P(S and B) or P(S given B)
2) Is the question asking for P( S and B)?
3) How do I utilize the P(S) probability in the calculations?

Thanks !
 
1. You didn't. You are told that 60% of smokers get sick; that is, P(B | S) = 0.60: given (i.e. if) someone smokes, the probability he gets sick is 0.60.
2. Yes.
3. Point (1) will help!

Another issue is that the 24% applies to the population in general, not necessarily to "a certain office". That may be relevant to another question.
 
juniet0331 said:
I have been trying to figure this question out and I may be overthinking it. I could really use some help and guidance.
Suppose that 24% of people suffer an acute breathing illness such as COPD by age 65. Research shows that 60% of smokers and 15% of non-smokers suffer from an acute breathing illness by age 65. In a certain office, 20% of the staff smokes cigarettes.
Let S = person smokes
B = person suffers acute breathing illness by age 65
1. Find the probability that a person in this office both smokes and will suffer an acute breathing illness by age 65.
I worked it out first that
P(B) =.24, P(S given B) = .60 and P(S) =.20
Issues:
2) Is the question asking for P( S and B)?
Click to expand...
Do you understand that
\(\displaystyle \mathscr{P}r(A\cap B)=\mathscr{P}r(A| B)\mathscr{P}r(B)=\mathscr{P}r(B| A)\mathscr{P}r(A)~?\)
 
@Dr.Peterson thank you ! That makes sense. Its the identification of P (B given S) versus P(S given B) that gets confusing.
@pka I do .
In this case to find P(S and B) I would do
P (S and B) = P (B given that S) * P(S)

There are actually 2 subquestions but I wanted to see if by working through question 1 I could figure out the other two.



2. Find the probability that a person in this office either smokes or will suffer an acute breathing illness by age 65.

3. Suppose that we know that someone in this office suffers from an acute breathing illness. What is the probability that they are a smoker?
 
Give it a try. I think you know what you need for each of these.

Now, I mentioned that the 24% doesn't necessarily apply to this office. But I just checked the implications of the 15%, which you didn't use, and it implies the 24%, so there is no contradiction in using it for part 2. I'm not sure why they include what turns out to be redundant information, or whether you are expected to do what I did (which I can show you after you finish, if you want).
 
I tried working out the other 2 problems
1. P(S or B) = P(S) + P(B) - P(S and B)
(. 20) *(. 20) - (.12, the answer from previous question)

2. P(S given B) = P (S and B) / P(B)
= (. 12) /(.24)

Comments or suggestions would be helpful again!
Thanks.
 
juniet0331 said:
I tried working out the other 2 problems
2. P(S or B) = P(S) + P(B) - P(S and B)
(. 20) *(. 20) - (.12, the answer from previous question)

3. P(S given B) = P (S and B) / P(B)
= (. 12) /(.24)

Comments or suggestions would be helpful again!
Thanks.
Click to expand...
1. Why do you put (. 20) *(. 20) for P(S) + P(B)? That's a wrong number and a wrong operation.

2. Good. Just carry out the division.
 
@Dr.Peterson i meant to type " + (. 24)" for " + P(B)".
I was typing it out on my phone and must have mistyped.

Would you happen to have any tips for being able to distinguish conditional probabilities in word problems when it's not clearly stated with "given"? Like the P(B given S) in this problem.

Thanks for the help!
 
I just paraphrase it, sometimes in steps:
  • Research shows that 60% of smokers and 15% of non-smokers suffer from an acute breathing illness by age 65.
  • 60% of smokers suffer from an acute breathing illness.
  • 60% of S are B.
  • Of S, 60% are B.
  • Given S, P(B) is 60%
  • P(B | S) = 60%
In particular, I focus on what group is the "whole" for the percentage. That's what's given.

Also, I often make tables, as I did here. In this case, rows were S and S' (not S), and columns were B and B'. Then I can think, out of the whole column labeled S, 60% are in the row B. That helps to focus on the whole and the part.

That table is also where I worked out the implication of 15% of non-smokers being B.
 
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