I'll denote

P(p) = prior probability of someone being contacted personally

P(ph) = prior probability of someone being contacted by phone

P(l) = prior probability of someone being contacted by letter

And

P(c) = probability of collecting an overdue ammount

We know:

P(p) = 0.7

P(ph) = 0.2

P(l) = 0.1

P(c | p) = 0.75

P(c | ph) = 0.60

P(c | l) = 0.65

And we want to find the posterior probabilities:

p(p | c) = ?

p(ph | c) = ?

p(l | c) = ?

From Bayes' Theorem we have:

p(p | c) = p(c | p) * p(p) / p(c) (1)

p(ph | c) = p(c | ph) * p(ph) / p(c) (2)

p(l | c) = p(c | l) * p(l) / p(c) (3)

All we have to compute is p(c), which is:

p(c) = p(c & p) + p(c & ph) + p(c & l) = p(c | p) * p(p) + p(c | ph) * p(ph) + p(c | l) * p(l)

Now we have all values and can substitute in (1), (2), (3).

EDIT: Bayes' theorem the other way around =X