Exercise 21 Page 698

We are told that the express bus arrives at our neighborhood between 7:30 and 7:36AM. The local bus arrives between 7:30 and 7:40AM. We want to find the probability that we will miss both buses if we arrive at the bus stop at 7:33AM. To do so we can use a probability tree diagram. Let's name our events and make a tree diagram.

Let's start by finding the probability of event E. We know that the express bus can arrive between 7:30 and 7:36AM, so there are 6 minutes in which the bus can arrive. We arrive at the bus stop at 7:33AM, so if the bus arrives in the first 3 minutes we will miss it.

P(E)=3/6=1/2 The local bus can arrive between 7:30 and 7:40AM, so there are 10 minutes in which the bus can arrive. Since we arrive at the bus stop at 7:33AM, we can miss the bus if it arrives during the first 3 minutes. P(L)=3/10 We can now add the obtained probabilities to our diagram.

We could find the other probabilities using the formula for the probability of a complement, but note that we do not need them in this case. We want to find the probability that we missed both buses, so we are only interested in the top branch of our diagram. To do so we need to multiply the probabilities on the top branch. Let's do it!

P(E and L)=P(E) * P(L|E)

SubstituteII

P(E)= 1/2, P(L|E)= 3/10

P(E and L)= 1/2* 3/10

Multiply

Multiply

P(E and L)=3/20

The probability that we missed both the express bus and the local bus is 320.