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

Answer:

$1 - p$

Step-by-step 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.