the risk of getting lung cancer, math question?

Since learning probability starts with experimentation, I hope you imagined or will imagine 1000 people of which 100 have lung cancer and go from there before "solving" the problem the textbook way.

As far as algebra goes, when we choose a random person, the probability that they are a

smoker: P(S) = 1/8

non-smoker: P(N) = 7/8

When you take a patient with lung cancer, the probability that it's a smoker is 0.9

P( S | L ) = 0.9

We're interested in the other perspective, the probability that a smoker develops lung cancer, i.e. P( L | S )

Bayes' formula says:

P( L|S ) = P( S|L ) * P(L) / P(S)

and

P( L|N ) = P( N|L ) * P(L) / P(N) 

Now, we can't calculate those themselves because we don't know how frequent lung cancer is P(L). However their ratio is

P( L|S ) / P( L|N ) = P( S|L ) / P( N|L ) * P(N) / P(S)

= 0.9 / 0.1 * (7/8) / (1/8)

= 9*7 = 63