Probability of Child Contracting Waterpokken

[discy]

Homework Statement

In a population of children 60% are vaccinated against the 'waterpokken'. The probablilities of contracting 'waterpokking' are 1/1000 if the child is vaccinated and 1/100 if not.

a: Find the probability that a child selected at random will contract 'waterpokken'.

Homework Equations

Bayes Theorem: P(A|B) = P(A/\B) / P(B)
Formula sheet? View attachment stat_formulas.pdf

The Attempt at a Solution

Known values: P(V) = 0.6 | P(⌐V) = 0.4 | P(W|V) = 0.001 | P(W|⌐V) = 0.01

a: Find the probability that a child selected at random will contract 'waterpokken'.

1. The question is: what is P(W)?
   From Bayes Theorem I conclude: P(W) = P(V/\W) / P(V|W)
   Both P(V/\W) & P(V|W) are unkown at this stage.
2. P(V/\W)?
   Fill in known values in bayes theorem:
   P(W|V) = P(W/\V) / P(V) = 0.001 = P(W/\V) / 0.6
   So: P(V/\W) = P(W/\V) = P(V) * P(W|V) = 0.6 * 0.001 = 0.0006
   Until here I get it! But now...
3. P(V|W)?
   P(V|W) = P(V/\W) / P(W) but I don't know P(W) :S?
   
   Answer sheet solution:
   P(W/\⌐V) = P(W|⌐V) = 0.01 * 0.4 = 0.004
   P(W) = P(W/\V) + P(W/\⌐V) = 0.0006 + 0.004 = 0.0046

1. What rules are applied in the third section? Could someone explain to me how they conclude to
P(W/\⌐V) = P(W|⌐V) and P(W) = P(W/\V) + P(W/\⌐V) ?

Maybe it's part of the formulas at the bottom of the first page of my formula sheet View attachment stat_formulas.pdf

2. Is there a way I could have known that I would need to work to that solution. Because I concluded I needed to work towards P(W) = P(V/\W) / P(V|W) but that plainly doesn't work..:(

 

[discy]

is my question not clear enough?:( The exam is tomorrow (tuesday).

 

[LCKurtz]

It doesn't require Bayes theorem. Just plug the numbers into

$$P(W) = P(W|V)P(V) + P(W|\overline V)P(\overline V)$$

 

[HallsofIvy]

I always prefer to work with integers! Suppose there are 10000 children. 6000 of them are vaccinated, 4000 are not. Of the 6000 who are vaccinated 6 of them contract the disease, 5994. Of the 4000 who are not vaccinated, 40 contract the disease.