Independent Events - Misconception

Given that P(Q) = q and P(R)= r , express P(Q or R) in terms of q and r when Q and R are independent


I know this means that Q and R affect one another's probabilities. I therefore understand that Q and R must intercept and the P(Q n R ) > 0 , assuming q =/= 0 and r=/=0 .

But surely P(Q or R) = q + r ? On the mark scheme they have P(Q or R) = q + r - qr which makes no sense as to why they'd subtract the intercepts.

Thanks.

Given that P(Q) = q and P(R)= r , express P(Q or R) in terms of q and r when Q and R are independent


I know this means that Q and R affect one another's probabilities. I therefore understand that Q and R must intercept and the P(Q n R ) > 0 , assuming q =/= 0 and r=/=0 .

But surely P(Q or R) = q + r ? On the mark scheme they have P(Q or R) = q + r - qr which makes no sense as to why they'd subtract the intercepts.

Thanks.

because the two events are independent, the chances of Q and R happening (where the venn diagram would overlap) is just the two probababilities multiplied. (e..g if Q was 0.3, R was 0.4, the chance of both would be 0.12).

Assuming Q and R are both not equal to 0, then there is going to be some overlap between the two events. For example if Q was the chance of it raining, and R was the chance of you seeing a yellow car on your way to work, these two events don't depend on each other (you're no more likely to see a yellow car whether it's raining or not), but there will be days where both occur. If you just counted P(Q or R) as q+r, you'd be counting the days where both Q and R happened twice, so you need to subtract the probability of them both happening from q+r, in order for the times where both Q and R occurred to only be counted once.

I don't know if this helps at all.

Are you familiar with the general result

$P(A\cup B) = P(A) + P(B) - P(A\cap B)$

If so, then the result follows immediately since in the case when A,B are independent we have

$P(A\cap B) = P(A)P(B)$

Where does that general rule come from though ?

Draw yourself a venn diagram.

The event (A u B) is when the circles are shaded in ONCE.

If you shade in the entire event A with one colour, and then the event B with another colour, you should end up shading in the intersection twice.

This is why you need to adjust and subtract one shading of the intersection so that everything is now with one layer of shading.