The Cigarette Question

[mrahroy]

The sample includes 30% of women and 70% of men. It is known that among them 10% of women and 60% of men smoke. Person X is known to be a non-smoker. Find the probability that X is male.

 

[Deleted member 4993]

The sample includes 30% of women and 70% of men. It is known that among them 10% of women and 60% of men smoke. Person X is known to be a non-smoker. Find the probability that X is male.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.

 

[Steven G]

You need to know exactly what the event that you want the probability of before continuing.

You want the probability that .

You will get another hint from me after you answer my question above or show some attempt in solving this problem.

 

[mrahroy]

You need to know exactly what the event that you want the probability of before continuing.

You want the probability that .

You will get another hint from me after you answer my question above or show some attempt in solving this problem.

I have calculated non-smokers as fol:- 60% are smokers so 40% are non-smoker males. 40% of 70 men make 28. And 10% of females are smokers. so 90% of 30 females make 27. Total non-smokers becomes 28+27 = 55. Now, to find the probability of X to be a male as well as a non-smoker:-
Probability of non-smoker X = 28/70 ?
or should it be 28/100?
Or even 28/55?

 

[Dr.Peterson]

The sample includes 30% of women and 70% of men. It is known that among them 10% of women and 60% of men smoke. Person X is known to be a non-smoker. Find the probability that X is male.

I have calculated non-smokers as fol:- 60% are smokers so 40% are non-smoker males. 40% of 70 men make 28. And 10% of females are smokers. so 90% of 30 females make 27. Total non-smokers becomes 28+27 = 55. Now, to find the probability of X to be a male as well as a non-smoker:-
Probability of non-smoker X = 28/70 ?
or should it be 28/100?
Or even 28/55?

You want the probability that X is a male GIVEN that X is a non-smoker. So the numerator is the number of male non-smokers, and the denominator is the number of non-smokers.

 

[BigBeachBanana]

You are given: $$\text{Pr(Male) = .30 , Pr(Female)=.70,Pr(Smoker|Male) = .60, Pr(Smoker|Female) = .10}$$The goal is to find:$$\text{Pr(Male | Smoker)}$$Using conditional probability (aka Bayesian's Probability) we have: $$\text{Pr(Male | Smoker)} = \frac{\text{Pr(Smoker} \cap \text{Male)}}{\text{Pr(Smoker)}}$$First, focus on the numerator. Again, using conditional probability, it can be rewritten as: $$\text{Pr(Male}\cap \text{Smoker}) = \text{Pr(Smoker | Male)*Pr(Male)}$$Next, the denominator. Using the Law of Total Probability & conditional probability: $$\text{Pr(Smoker)} = \text{Pr(Smoker} \cap \text{Male}) + \text{Pr(Smoker} \cap \text{Female}) = \text{Pr(Smoker|Male)*Pr(Male) + Pr(Smoker|Female)*Pr(Female)}$$Putting it all together:
$$\text{Pr(Male | Smoker)} =\frac{\text{Pr(Smoker | Male)*Pr(Male)}}{\text{Pr(Smoker|Male)*Pr(Male) + Pr(Smoker|Female)*Pr(Female)}}$$Hopes this help you understand Bayesian's Probability more concretely.

 

[Dr.Peterson]

You are given: $$\text{Pr(Male) = .30 , Pr(Female)=.70,Pr(Smoker|Male) = .60, Pr(Smoker|Female) = .10}$$The goal is to find:$$\text{Pr(Male | Smoker)}$$Using conditional probability (aka Bayesian's Probability) we have: $$\text{Pr(Male | Smoker)} = \frac{\text{Pr(Smoker} \cap \text{Male)}}{\text{Pr(Smoker)}}$$First, focus on the numerator. Again, using conditional probability, it can be rewritten as: $$\text{Pr(Male}\cap \text{Smoker}) = \text{Pr(Smoker | Male)*Pr(Male)}$$Next, the denominator. Using the Law of Total Probability & conditional probability: $$\text{Pr(Smoker)} = \text{Pr(Smoker} \cap \text{Male}) + \text{Pr(Smoker} \cap \text{Female}) = \text{Pr(Smoker|Male)*Pr(Male) + Pr(Smoker|Female)*Pr(Female)}$$Putting it all together:
$$\text{Pr(Male | Smoker)} =\frac{\text{Pr(Smoker | Male)*Pr(Male)}}{\text{Pr(Smoker|Male)*Pr(Male) + Pr(Smoker|Female)*Pr(Female)}}$$Hopes this help you understand Bayesian's Probability more concretely.

@mrahroy, You realize, I hope, that this answers the wrong question, though it shows all the details of the method for solving the right question:

The sample includes 30% of women and 70% of men. It is known that among them 10% of women and 60% of men smoke. Person X is known to be a non-smoker. Find the probability that X is male.

Your work had the right numerator, and just need to decide on the denominator.