simple distribution

the number of eggs laid by a kind of rare bird in its breeding season may be considered a random variable X with the probability distribution given by P(X=0) = 0.2
P(X=1) = 0.24 P(X=2) = 0.32 P(X=3) = 0.24

the expert observes that the probability of the survival of a baby bird is dependent on the number of eggs x laid per pair of adult birds.

1 egg : survival = 0.9, 2 eggs : survival = 0.7, 3 eggs : survival = 0.4

a) let W be the number of baby birds surviving per pair of adult birds. then W = 0 denotes that a pair of adult birds do not lay egg, or the eggs laid are all dead. Find the probability distribution of W.

Answers : P(W=0) = 0.30464, P(W=1) = 0.45408, P(W=2) = 0.22592,
P(W=3) = 0.01536


Please explain how you count the probabilities, thanks.