# Question: Events Q and R

## Comment on Events Q and R

### If 0.45 is the complement,

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! ### Great question!!

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

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? ### We can still interpret 'and'

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: https://www.greenlighttestprep.com/module/gre-probability/video/752

### would you recommend more

would you recommend more practice resources about probability. ### We have plenty of links to

We have plenty of links to additional practice questions in the Related Resources boxes of most lessons (e.g., https://www.greenlighttestprep.com/module/gre-probability/video/752).

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

Hi Brent, Can you please tell us how to prioritize the subjects under quants?
if one wants to score 167+ ### If you want a 167+ score,

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

### Hello Brent,

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 ### I'm sure where you got the

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

Why P(R|Q) is 1 or less than 1?
I have doubts. Kindly explain ### The greatest possible value

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 https://www.greenlighttestprep.com/module/gre-probability/video/742)

Does that help?

Cheers,
Brent