Question

Conditional probability or intersection?

A health study tracked a group of persons for five years. At the beginning of the study,
20% were classified as heavy smokers, 30% as light smokers, and 50% as nonsmokers.
Results of the study showed that light smokers were twice as likely as nonsmokers to
die during the five-year study, but only half as likely as heavy smokers. A randomly
selected participant from the study died over the five-year period. Calculate the
probability that the participant was a heavy smoker.

Answer 1

If I let H = heavy smoker, L = light smoker, N = nonsmoker, D = died over 5-year study. I know I have
P(H) = 0.2
P(L) = 0.3
P(N) = 0.5

But for the next probabilities, I don't know if I should be writing:
P(D∩L) = 2 P(D∩N) = 1/2 P(D∩H)
or:
P(D|L) = 2 P(D|N) = 1/2 P(D|H)

Is there a way to know if it's a conditional probability, or an intersection?