There are two traffic lights on the route used by a certain

There are two traffic lights on the route used by a certain individual to go from home to work. Let E denote the event that the individual must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.2, P(F) = 0.4, and P(E F) = 0.13.

(a) What is the probability that the individual must stop at at least one light; that is, what is the probability of the event P(E F)?


(b) What is the probability that the individual doesn\'t have to stop at either light?


(c) What is the probability that the individual must stop at exactly one of the two lights?


(d) What is the probability that the individual must stop just at the first light? (Hint: How is the probability of this event related to P(E) and P(E F)? A Venn diagram might help.)

Solution

a) BY UNIVERSAL IDENTITY
P(E F) = P(E) + P(F) - P(E F)
= 0.2 + 0.4 - 0.13
= 0.47

b) COMPLEMENT OF UNION OF BOTH
1- P(E F) = 1- 0.47 = 0.53

c) IN THIS WE HAVE TO COUNT 1 IN THE INTERSECTION OF BOTH CASE WISE
P( E F\') + P( E\' F)
= (0.2)(1-0.4) + (1-0.2)(0.4)
= 0.44

d) WE HAVE TO COUNT THE PROBABILITY OF P(E)
P(E) = P( E F) + P ( E F\')
= (0.2)(0.4) + (0.2)(1-0.4)
= 0.2