On his drive to work, Leo listens to one of three radio

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On his drive to work, Leo listens to one of three radio stations A, B or C. He first turns to A. If A is playing a song he likes, he listens to it; if not, he turns it to B. If B is playing a song he likes, he listens to it; if not, he turns it to C. If C is playing a song he likes, he listens to it; if not, he turns off the radio. For each station, the probability is 0.30 that at any given moment the station is playing a song Leo likes. On his drive to work, what is the probability that Leo will hear a song he likes?

A. 0.027
B. 0.090
C. 0.417
D. 0.657
E. 0.900

OA D

The probability that Leo will hear a song he likes on the way to work is the probability he will not turn off his radio. That is, either station A will be on for the entire trip, or station B or C will be on by the end of the trip.

The probability that station A will be on for the entire trip is 0.3.

Station B will be on by the end of the trip if station A did not play a song he likes AND station B did play a song he likes. The probability is 0.7 x 0.3 = 0.21.

Station C will be on by the end of the trip if station A did not play a song he likes AND station B did not play a song he likes AND station C did play a song he likes. The probability is 0.7 x 0.7 x 0.3 = 0.147.

Since these events are mutually exclusive, we add their probabilities, so the probability that a station will be on by the end of the trip is 0.3 + 0.21 + 0.147 = 0.657.

Alternate Solution:

We can use the formula:

P(Leo hears a song he likes) + P(Leo does not hear a song he likes) = 1

Notice that since the probability of any station playing a song Leo likes is 0.3, the probability that any station playing a song Leo does not like is 0.7. The probability that none of the stations play a song Leo likes is 0.7 x 0.7 x 0.7 = 0.343. Thus, the probability that Leo hears a song he likes is 1 - 0.343 = 0.657.

Answer: D