00:01
It's given in this exercise that the probability that a motorist must stop at the first traffic signal is 0 .4, the probability that he must stop at the second is 0 .5, and the probability that he must stop at at least one is 0 .7.
00:17
So we have these three probabilities shown here, and for part a we're asked for the probability that he must stop at both signals.
00:28
So that is, we want to find the probability of event a and event b.
00:33
So to solve this we can apply the sum rule for probability.
00:42
This rule tells us that the probability of event a or b is equal to the sum of their individual probabilities minus the probability of their intersection.
01:01
So if we fill in the values we end up with the probability of a and b is equal to the probability of a plus the probability of b minus the probability of a or b, and this comes out to 0 .2.
01:28
For part b we want the probability that he stops at the first signal but not the second.
01:36
So this is the probability of a and b complement.
01:43
So using the multiplication rule we can write this as the probability of b complement given a times the probability of a and we can look at this probability.
02:01
The complement of b complement given a is b given a so we can express this as 1 minus the probability of b given a.
02:17
B given a is the complement of b complement given a.
02:21
So we write these probabilities are equal like this.
02:27
Put brackets around that times the probability of a.
02:36
And then for the second term inside the brackets, use the multiplication rule, write this as the probability of b and a divided by the probability of a.
02:55
So we have this from part a, that's 0 .2.
02:59
So we have everything we need to solve this.
03:12
And this comes out to 0 .2...