Leo listening to the radio :-)

[Steph]

Hi all,
sigh.. here is my least favorite type of problem :-( help please! thanks,

On his drive to work, Leo listens to one of 3 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 to B. If B is playing a song that he likes, he listens to it; if not, he turns to C. If C is playing a song he likes, he listens; if not, he turns off the radio. For each station, the probability is 0.3 that at any given moment the station is playing a song Leo likes. On his drive to work, what is the probability that Leo ill hear a song he likes?

a. 0.027
b. 0.09
c. 0.417
d. 0.657
e. 0.9

The correct answer to this prob is D

[Guest]

Pretty St Forward:

Will get his song on A: 0.3 -1
Will not get his song on A: 0.7

Will get his song on B: 0.7 * 0.3 (i.e. not get the song on A * Get his song on B) -2
Will not get his song on (A and B): 0.7 * 0.7

Will get his song on C: 0.7 * 0.7 * 0.3 (i.e. not get the song on A * Not get his song on B * Get his song on C) -3

Since 1 2 and 3 are mutually exclusive:

It will be: 1 + 2 + 3 Will give you the answer.

[RonPurewal]

Pretty St Forward:

Will get his song on A: 0.3 -1
Will not get his song on A: 0.7

Will get his song on B: 0.7 * 0.3 (i.e. not get the song on A * Get his song on B) -2
Will not get his song on (A and B): 0.7 * 0.7

Will get his song on C: 0.7 * 0.7 * 0.3 (i.e. not get the song on A * Not get his song on B * Get his song on C) -3

Since 1 2 and 3 are mutually exclusive:

It will be: 1 + 2 + 3 Will give you the answer.

correct.

just to clarify, the answer that comes from this method (which is the correct answer, btw) is 0.3 + (0.7)(0.3) + (0.7)(0.7)(0.3), or 0.657.

--

you could also use the complementary event to solve this problem with fewer steps.
the opposite of the desired event is "leo doesn't hear a song he likes". that's an easier event to consider, because it's not a compound event; it's only one possibility, namely, "no" on all three. that probability is (0.7)(0.7)(0.7) = 0.7^3, or 0.343.
therefore, the desired probability is 1 - 0.343, or 0.657.