Highest possible probability for intersection of the events…

I had this question

“Suppose that you select a student at random from a particular college, and you record their gender and whether they prefer to use a PC or a Macintosh for their computing. Suppose that following probability P(student uses mac) = .3 , P(student uses pc) = .7, P(student is a male) = .4, and P(student is a female) = .6. Answer the following…

a.) What is the probability student is male and student is mac user if the events are independent.

b.) what is the probability student is male and student is mac user if the events are mutually exclusive

c.) what is the highest possible probability for the intersection of the events student is male and student is mac user

My answers:

a.) P( mac intersects male) = .4*.3 = .12 (correct)

b.) P(mac intersections male) = 0 (correct)

c.) independent (wrong)

The reason I said independent is because it is obviously higher than mutually exclusive. Now two events with probability greater than zero that are mutually exclusive cannot be independent. So, that means if two events with non zero probabilities that are mutually exclusive are dependent. However, dependency of two events doesn’t guarantee disjointedness.

There is not enough information to calculate the P(male intersects mac) when male and mac are dependent but not disjoint. That would require the conditional probability of male given mac or mac given male. I am going to talk to the professor tomorrow about this (he asked me to see him, and asked that I retry the questions and then compare them to his) but I really dont know what I am doing wrong with part C. Unless I am just misunderstanding it completely.

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The highest possible probability is … a value. The answer will be a real number.
– Graham Kemp
Oct 21 at 1:59

  

 

Are you saying I should have put “.12” instead of “independent” ? My exact answer “3 choices: intersection when : independent (.12), disjoint (0), and dependent. There is not enough information to calculate the intersection of events when they’re dependent but not disjoint therefore the highest possible probability to the intersection is when the events are independent”
– Matthew Delengowski
Oct 21 at 2:03

  

 

There isn’t enough probability to calculate P(male student, mac user)P(\text{male student, mac user})—but they’re not asking you to calculate it; they want you to figure out what’s the greatest value it could take on. If you were to draw a Venn diagram of the two events, their intersection would be the joint event. Under what circumstances would that intersection be the largest possible?
– Brian Tung
Oct 21 at 2:03

  

 

@MatthewDelengowski: No, 0.120.12 is wrong also.
– Brian Tung
Oct 21 at 2:04

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1 Answer
1

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P(Mac)=0.3,P(Male)=0.4\mathbb P(\textrm{Mac})=0.3, \mathbb P(\textrm{Male})=0.4

Recall Venn Diagrams.   You have two circles of areas 0.30.3 and 0.40.4. What is the largest overlapping area you can make?

That gives you the least upper bound on the probability of the intersection.   It cannot be any greater than that, so it is the highest possible probability.

  

 

So when they’re on top of each other. The circle of one is completely inside the other.
– Matthew Delengowski
Oct 21 at 2:12

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So, the answer is .3
– Matthew Delengowski
Oct 21 at 2:14