"Randomly Playing Songs
Suppose a Spotify playlist you just created has 13 tracks. After listening to the playlist, you decide that you like 5 of the songs. The random feature on Spotify is set up to play each of the 13 songs once in a random order. Find the probability that among the first 4 songs played:


b. You like 3 of them;
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Verified step by step guidance

1. Identify the total number of songs in the playlist, which is 13, and the number of songs you like, which is 5.
2. Recognize that the problem involves selecting 4 songs from the 13 without replacement, and we want exactly 3 of these 4 songs to be ones you like.
3. Use the hypergeometric probability formula, which is appropriate for sampling without replacement. The formula is: \[ P(X = k) = \frac{\binom{K}{k} \times \binom{N-K}{n-k}}{\binom{N}{n}} \] where: - \(N\) is the total number of items (13 songs), - \(K\) is the number of success states in the population (5 liked songs), - \(n\) is the number of draws (4 songs played), - \(k\) is the number of observed successes (3 liked songs in the first 4).
4. Calculate each combination term: - \(\binom{5}{3}\): ways to choose 3 liked songs, - \(\binom{8}{1}\): ways to choose 1 song you do not like (since 13 - 5 = 8), - \(\binom{13}{4}\): total ways to choose any 4 songs from 13.
5. Substitute these values into the hypergeometric formula and simplify the fraction to find the probability that exactly 3 of the first 4 songs played are ones you like.

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population without replacement. In this problem, it helps calculate the chance of selecting exactly 3 liked songs out of the first 4 played, given 5 liked songs in a total of 13 tracks.

Combinatorics (Combinations)

Combinations count the number of ways to choose items from a set without regard to order. Here, combinations are used to determine how many ways to select liked and disliked songs from the playlist to form the first 4 songs played.

Probability of Events Without Replacement

When items are selected without replacement, each choice affects the next. This concept is crucial because the playlist plays songs in a random order without repeats, influencing the calculation of the probability that exactly 3 of the first 4 songs are liked.