Solving Club Membership Probability Problem

[chrisyuen]

Homework Statement

In a certain club, one-fifth of its members are smokers. One-sixth of its male members are smokers. Among the non-smoking members, one-eighth are female. A member is randomly chosen.

(a) What is the probability that this member is male;
(b) If this member is female, what is the probability that she is a non-smoking member.

(Answers:
(a) 21/25
(b) 5/8)

Homework Equations

Probability Formulae

The Attempt at a Solution

One of my friends told me the followings in order to get the final answer correct.

However, I don't know how can he get them.

(a)
(1 - 1/5) * (1 - 1/8) / (1 - 1/6)
= 7/10 / (5/6)
= 21/25
(b)
(1 - 1/5) * 1/8 / (1 - 21/25)
= 1/10 / (4/25)
= 1/10 * 25/4
= 5/8

I tried to write a tree diagram as follows:

S = 1/5
NS = 4/5

S.M = 1/6
S.F = 5/6

NS.M = 7/8
NS.F = 1/8

S: Smoker; NS: Non-Smoker; M: Male; F: Female

Did my tree diagram correct?

My attempt on part (a):
P(M)
= P(M|S)P(S) + P(M|NS)P(NS)
= 1/6 * 1/5 + 7/8 * 4/5
= 11/15

My attempt on part (b):
P(NS|F)
= P(NS and F) / (P(F|S)P(S) + P(F|NS)P(NS))
= 4/5 * 1/8 / (1/5 * 5/6 + 4/5 * 1/8)
= 3/8

But my answers are not correct.

Can anyone tell me how to solve this question?

Thank you very much!

 

[Focus]

P(S|M) is not the same as P(M|S) this was your error on both the parts. Do you know the formula to obtain P(M|S) from P(S|M)?

 

[chrisyuen]

P(S|M) is not the same as P(M|S) this was your error on both the parts. Do you know the formula to obtain P(M|S) from P(S|M)?

P(S|M)
= P(S and M) / P(M)
= P(M|S)P(S) / P(M)?

Did my tree diagram correct?

Thank you very much!

 

[Focus]

P(S|M)
= P(S and M) / P(M)
= P(M|S)P(S) / P(M)?

Did my tree diagram correct?

Thank you very much!

I don't know about tree diagrams but one of them looks false. Try to just write it out as P(A|B). "One-sixth of its male members are smokers", what are you given?

As an example, 2/5 of the women population read Heat magazine, so P(reading heat| it's a woman)=2/5. The information given to you is that it is a woman, restated it says given that the person is a woman, the probability of the person reading Heat is 2/5.

Your formula is correct (Bayesian formula). Try to rethink about what the question says. I hope that example helps. If not let me know.

 

[chrisyuen]

P(NS|M)
= P(M|NS) P(NS) / P(M)

P(M)
= P(M|NS) P(NS) / P(NS|M)
= P(M|NS) P(NS) / (1 - P(S|M))
= 7/8 x 4/5 / (1 - 1/6)
= 21/25

P(NS|F)
= P(F|NS) P(NS) / P(F)
= P(F|NS) P(NS) / (1 - P(M))
= 1/8 x 4/5 / (1 - 21/25)
= 5/8

Am I right?

 

[Focus]

P(NS|M)
= P(M|NS) P(NS) / P(M)

P(M)
= P(M|NS) P(NS) / P(NS|M)
= P(M|NS) P(NS) / (1 - P(S|M))
= 7/8 x 4/5 / (1 - 1/6)
= 21/25

P(NS|F)
= P(F|NS) P(NS) / P(F)
= P(F|NS) P(NS) / (1 - P(M))
= 1/8 x 4/5 / (1 - 21/25)
= 5/8

Am I right?

Yep looks fine to me

 

[chrisyuen]

Yep looks fine to me

Thanks for your helps!