I have a question about a problem titled “A is the probability it will rain tomorrow and B…” trying to send you a screenshot.

My question is, should not these be mutually exclusive?

How can it be I am sleeping and awake at the same time,

or I am eating and not eating at the same time.

If it rains, could it not rain at the same time or same day?

Or, if it rains any time of the day, it means 100% true that it rained. I thought GRE could use this sense to trap us.

I mean, I think GRE expects us to guess this intuition.

Again on the other hand, how we can be sure rain and not rain makes up 1?

For example, if it is said that from the sack probability of picking red ball is 30% and blue is 20%, that means there are other balls too.

But in this case, there is no other way, either it will rain or not, that is why it makes 1 when adding up together.

And if it is true, these two cases should be mutually exclusive and the result should be 0.

I do not challenge Greg, I want to learn what should I guess in this case? Can I bring outside knowledge which is a kind of fact?

As you know this event is mutually exclusive like a coin flip that is you either get head or tail

there is 50-50 probability that you will get something but not head and tail at the same time.

like you said there will be rain or not rain that is impossible .

There is only one possible event that is rain or no rain (don’t take it as natural event or phenomena . In nature thing work differently .)

But the essence of this question is that this event is mutually exclusive and only one event occur at time

total event is 2

so probabilty of rain 1/2

probability of not rain 1/2

And mot importantly in Probabily AxB is adding of both event so 1/2+1/2 gives the result .

Hope it make clear to you.

Hi Puerile, thanks a lot for your kind reply.

I am clear about your explanation of first half, but still in confusion for the last line.

How it could be A X B equals 1/2 + 1/2= 1?

I know how it did make 1 when sum up. But it is not possible for happening of both the head and tail from one toss?

Thanks

To understand that you need to read the last part of questions again and again .

You know what is maximum probability of any mutually exclusive events that is 1

Thanks for your reply.

Let’s take an example, a fair coin has a possibility of showing head, denoted by A, 1/2 and tail, denoted by B, 1/2.

What is the possibility of AXB = 1/2 * 1/2 = 1/4.

But is it possible for a coin to show both head and tail (AXB).

It is possible if it is said the coin was tossed for two times or more and then what is the possibility of having one head and one tail or any other combination, then we will do 1/2 X 1/2 X …

Hi

This case is indeed similar to flipping a coin. The events are mutually exclusive since the occurrence of one event(rain) prevents the occurrence of other(not rain) and in case of mutually exclusive the sum of probabilities of individual events is always equal to 1. Here there are only two events rain and no rain. So sum of probabilities( probability of rain-A and no rain-B) is 1.

So A+B=1

B = A-1

A \times B = (A-1)A = A \times A - A

for which the maximum value occurs when A=1/2 and B=1/2

so A \times B = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}