There are three general locations that a taxi can go to the

There are three general locations that a taxi can go to: the airport, downtown, and elsewhere. When a taxi driver starts in the downtown location, there is a 0.40 chance that his first call will take him to the airport and a 0.40 chance of going to another downtown location. Once a taxi is at the airport, there is a 0.80 probability that the next fare will take him downtown and a 0.20 chance of going elsewhere. The probability of a call from anywhere except downtown taking him to the airport is 0.20. Therefore, the probability that the taxi is at the airport when the third call arrives after going on shift is 0.20. True or False?

Solution

Probability of going to elsewhere on the first call is 1-0.4-0.4=0.2.

First route:

taxi go to downtown on first call and then airport on second call. The probability of this route is 0.4*0.4=0.16

Second route:

taxi go to elsewhere on first call and then airport on second call. The probability of this route is 0.2*0.2=0.04

In both possbile routes taxi will be at airport on third call. Both route are mutually exclusive. So the probability that taxi is at airport when third call arrives is

0.16+0.04=0.20

So answer is true.