How do I calculate this easy probability example

Hey,
I've got a seemingly easy probability question that I can't find the answer to..

There are two people that either vote - or not. Probability of any one of them voting is P.
(Independent events)

I want to find the probability that neither of them vote and find two solutions to this. The second one is right - but why is the first one wrong?

I)
probability that one of them votes is P
probability that both vote is P²
probability that neither vote must therefore be 1 - P²

II)
True seems to be:
probability that one of them votes is P
probability that he doesn't vote is 1-P
probability that neither vote is (1-P)x(1-P) = (1-P)²

1-P² is of course not equal to (1-P)²

Can you guys help my mind? :-)

Thank you all!!

probability thinking can be tricky ... here is why I is wrong while II is correct:

the probability that neither vote equals the probability that BOTH don't vote, i.e. (1-p)^2.

Your first answer (I) is the probability that ANY one of them didn't vote, meaning it includes not only the probability that they BOTH didn't vote, but also the probabilities that either one of them didn't vote while the other one did.

You can see this by looking at all the cases:

- both vote = p^2
- both don't vote (neither votes) = (1-p)^2 = 1 - 2p + p^2
- either one votes = 2*p(1-p) = 2p - 2p^2

let's sum them all up to check:

p^2 + 1-2p+p^2 + 2p-2p^2 = 1 - which means we took into account all possible outcomes

Its actually a binomial probability.


The probability that neither of them vote is the SAME as the probability that both of them vote.

>"How do I calculate this easy probability example"

If it's easy then I don't understand the problem! Maybe you should consider opening up your textbook!

Prob( neither vote) = 1 - Prob(at least one of them voting), not 1 - Prob (both vote) as you assert in 1)