Looking at slide 11, Example 1.10 from:
http://www-users.aston.ac.uk/~cornford/probmod/ProbMod310810_Ch1.pdf
Luke has been told he’s lucky and has won a prize in the lottery. There are 5 prizes available of value £10, £100, £1000, £10000, £1000000. The prior probabilities of winning these 5 prizes are $p_1, > p_2, p_3, p_4, p_5$, with $p_0$ being the prior probability of winning no prize.
Luke asks eagerly ‘Did I win £1000000?!’. ‘I’m afraid not sir’, is the response of the lottery phone operator. ‘Did I win £10000?!’ asks Luke. ‘Again, I’m afraid not sir’.
What is the probability that Luke has won £1000?
I assume Bayes' formula would work here: $P(x\mid j) = \dfrac{P(j\mid x)P(j) }{ P(x)}$
If so, I see how the denominator in their answer:
$P(w \neq 5, w \neq 4, w \neq 0)$ comes from
but I am not able to figure out how they got the numerator: $p(w=3, w \neq 5, w \neq 4, w \neq 0)$
any help is appreciated.