Question: Events Q and R

Comment on Events Q and R

If 0.45 is the complement, doesn't it represent "the probability that both won't occur" (i.e, that either Q will occur, R will occur, or neither will occur)?

And doesn't it intuitively follow that the probability of those three events is going to be greater than just the probability of Q occurring alone?

That was my (non-arithmetic) logic. Why doesn't it work?

Thank you!
greenlight-admin's picture

Great question!!

Please note that P(Q occurs) is NOT the same as P(ONLY Q occurs). There are 4 possible outcomes:
1) only Q occurs
2) only R occurs
3) neither Q not R occurs
4) Q and R both occur

In outcomes 1 and 4, Q occurs, so our calculations must include these two outcomes.

You're right about the complement. That is P(only Q occurs OR only R occurs OR neither Q not R occurs) = 0.45

In other words, P(only Q occurs) + P(only R occurs) + P(neither Q not R occurs) = 0.45

So, P(only Q occurs) = some value between 0 and 0.45

We also know that P(both P and Q occur) = 0.55

So, P(Q occurs) = P(only Q occurs OR both P and Q occur)
= P(only Q occurs) + P(both P and Q occur)
= (some value between 0 and 0.45) + 0.55
= a value greater than or equal to 0.55

I think the question is incomplete. It should mention that the two events are independent in order to be able to interpret 'and' as multiply?
greenlight-admin's picture

We can still interpret 'and' as multiply. We just use the formula: P(A and B) = P(A) x P(B|A), where P(B|A) is the probability that event B occurs GIVEN THAT event A has occurred.

This formula can be applied to both independent and dependent events. For more on this, watch:

would you recommend more practice resources about probability.
greenlight-admin's picture

We have plenty of links to additional practice questions in the Related Resources boxes of most lessons (e.g.,

Having said that, please keep in mind that you won't see many probability questions on test day (perhaps 1 or 2 at most), so don't spend a disproportionate amount of time on probability if you have any weaknesses in the other (more often tested) topics such as algebra and statistics.

Hi Brent, Can you please tell us how to prioritize the subjects under quants?
if one wants to score 167+
greenlight-admin's picture

If you want a 167+ score, then EVERY quant topic is a priority.

Hello Brent,
I employed a different method. Please let me know if correct.

Step 1 : P(Q or R) = P(Q) - P(Q and R) + P(R) - P(Q and R)
Step 2 : P(Q or R) = P(Q) + P(R) - 2P(Q and R)
Step 3 : P(Q or R) = P(Q) + P(R) - 2*(0.55)
Step 4 : P(Q or R) - P(R) + 1.1 = P(Q)
Step 5 : Since P(Q or R) > P(R), therefore their subtraction should be positive.
Step 6 : Positive value + 1.1 should be greater than 0.55, therefore P(Q) is greater.

I know this is lengthy, but need a confirmation if I am right with the logic. Especially Steps 1 & 5
greenlight-admin's picture

I'm sure where you got the second P(Q and R)

It should just be P(Q or R) = P(Q) + P(R) - P(Q and R)

Why P(R|Q) is 1 or less than 1?
I have doubts. Kindly explain
greenlight-admin's picture

The greatest possible value for any probability is 1
If P(some event) = 1, then that event will definitely happen.

For example, let's say we randomly choose a number from the set {1, 2, 3, 4, 5}
P(the selected number is less than 100) = 1
In other words, we can be 100% certain that the selected number is less than 100.

In general, the probability of any event is greater than or equal to 0 and less than or equal to 1.

For more on this, start watching the following video at 0:25

Does that help?


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