# Mutually exclusive events and probabilities.

This is a question set in my maths class and I think it has a trick question. It's super simple, probably far too simple for this website, but here goes anyway.

Assume that we have two events, $$A$$ and $$B$$, that are independent and mutually exclusive. Assume further that we have $$P(A)=0.1$$ and $$P(B)=0.1$$. What is $$P(A \cap B)$$?

I am thinking the answer is $$0$$, as if the events are mutually exclusive they should never happen at the same time.

The reason I am stuck is the teacher has said the answer is $$0.1 \times 0.1$$.

Another small point:

Question

If I have a fair coin and I toss it 3 times then I think the sample space is?

{HHH, HHT, HTT, HTH, TTT, TTH, THH, THT}?

Am I correct and if not why not?

I do think they are correct but I would just like to double check?

• it is not possible for two events to be independent and mutually exclusive (excluding cases where one or the other of the events has probability $0$). – lulu Feb 25 '19 at 14:05
• what about if I removed the independent part from the question? – user22485 Feb 25 '19 at 14:05
• If you just assume they are mutually exclusive then, by definition, $P(A\cap B)=0$. If you just assume they are independent then $P(A\cap B)=P(A)\times P(B)$. – lulu Feb 25 '19 at 14:06
• perfect, I should speak to my teacher about this. – user22485 Feb 25 '19 at 14:07
• Yes, those are the possible outcomes from three tosses of a coin. – lulu Feb 25 '19 at 14:16

You are right. If two events are mutually exclusive, it means that they cannot happen at the same time (see here). This means that $$A\cap B=\varnothing$$ and $$P(A\cap B)=0$$.