Probability: One at a Time - Outcome A AND Outcome B = Multiply Probabilities
Bill's compact disc player randomly plays a song so that no song is repeated before the entire album is played. If Bill plays a disc with 14 songs, what are the chances that the third song he hears will be his favorite?
**Alternative method**:
A much easier thing to recognize is that Bill's favorite song is equally likely to be in any of the 14 positions (first through fourteenth). That means the probability is
>$$\displaystyle \text{Probability} = \frac{\text{# of positions you want}}{\text{# of total positions}} = \frac{1}{14}$$.
Incorrect.
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Incorrect.
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Incorrect.
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Incorrect.
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Correct.
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For the third song to be Bill's favorite, songs 1, 2, and 4 through 14 should not be Bill's favorite song. There are 13 unfavorable songs. The probability of that happening is
>$$\text{Probability} = \frac{13}{14}\times\frac{12}{13}\times\frac{1}{12}\times\frac{11}{11}
\times\frac{10}{10}\times\frac{9}{9}\times \ldots \times\frac{1}{1}=
\frac{1}{14}$$.
Note that from the fourth pick onwards, the probability of an unfavorable song being played is always 1, since the only favorite has already been played. Reduce the first three fractions:
>$$\text{Probability} = \frac{13}{14}\times\frac{12}{13}\times\frac{1}{12} = \frac{1}{14}$$.
$$\frac{1}{14}$$
$$\frac{1}{12}$$
$$\frac{1}{11}$$
$$\frac{3}{14}$$
$$\frac{1}{3}$$
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