Question

The probability that Paul wins in a raffle is given by the expression

p.

Write down an expression for the probability that Paul does not win.

Answer 1

`1 - p`

Explanation:

Consider the
`(\Omega, \mathcal{F}, \mathbb{P})` where
`\mathcal{F}` is sigma algebra and
`\mathbb{P}` is probabilistic measure. Denote
`A \subset \Omega` where Paul wins. By additivity of measure we know that

`\mathbb{P}(A) + \mathbb{P}(\Omega \setminus A) = \mathbb{P}(\Omega) = 1`.

So

`\mathbb{P}(\Omega \setminus A) = 1 - \mathbb{P}(A) = 1 - p`.

But
`\Omega \setminus A` is exactly the set where Paul does not win. Q.E.D.