Probability Using Venn Diagram

Question

Jill has applied for a job with each of two different companies A & B. Some probabilities are given in the Venn diagram (attached file), solve for P (A & B) and P (None).

Could someone please help me figure out the discrepancy between Venn Diagram solution vs probability solution while trying to solve for P (A & B) and P (None)? I'm getting different values for both if I try using those 2 different methods as shown by "??" in the uploaded file.

I was running into a similar issue while trying to solve  OG17 DS283 , so I made this simplified Q for understanding the basics.

Thanks!

IanStewart · Answer

The rule P(A and B) = P(A)*P(B) is only true if A and B are "independent". That is, it's only true if the outcome of A has no influence on whether B happens. That's sometimes true in real life -- if you roll a die and flip a coin, say, one event has no influence on the other. But it's sometimes not true -- the probability it rains in one city is probably very closely related to the probability it rains in a neighbouring city. In that case, you couldn't use the multiplication rule.

In your Venn diagram, your two events are not independent. If they were independent, then any time the applicant got job A, they'd still have a 50% chance of getting job B. But they don't - when they get job A, they only get job B 3/7 of the time (0.7 of the time they get A, and 0.3 of the time they get both). Since your events are not independent, you can't use the familiar multiplication laws in that situation, and you'd normally need to solve using another method, like the Venn diagram you've drawn.

Fortunately on the GMAT, in almost all questions, events are independent, so you will almost always be able to use the simple multiplication rules when you want the probability two events happen.

dabaobao · Answer

Thanks  IanStewart . In that case, would the independent events have a Venn diagram with two disjoint circles or overlapping lapping with extremely small overlap? My understanding is that they will have an extremely small overlap as shown in my uploaded file. Since the overlap for such events (i.e. my example on the right half of the page) is so small [Comparing P(A) = P(B) = 0.1 to P (A&B) = 0.1*0.1 = 0.01), they are called independent events even though they have a minor overlap.

Are events considered independent only when P(A&B) = P(A)P(B)?

Please correct me if my understanding is not fully correct. Thanks!

IanStewart · Answer

Yes, that is actually one way to define "independent" - two events are independent exactly when P(A and B) = P(A) * P(B).

If two independent events each have some possibility of occurring, then it will always be possible that both occur (i.e. P(A)*P(B) > 0 will always be true), so in a Venn diagram, the two circles would always overlap. That overlap can be big or small, depending on how big each probability is. If the probability is 1/10 that A happens, and 1/10 that B happens, then the overlap would be 1/100, so it's tiny because the probabilities are both so small. In the "only A happens" section of the Venn diagram you'd have 9/100, in the "only B happens" section you'd also have 9/100, and in the "neither A nor B happens" section you'd have 81/100.

But if the probabilities were both large, then the overlap would be large. If there's a 90% chance of rain in city X on any given day, and a 90% chance of rain in some distant city Y on any given day, and if those events are independent, the overlap would be 0.81, because it's very likely both events happen.

That said, you wouldn't generally use a Venn diagram to solve a problem with independent events - it might be useful to see how the Venn works just to help with conceptual understanding, but you'd usually just use the multiplication rules directly to solve an actual GMAT question.

Probability Using Venn Diagram

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