Answer:
- P(playlist has one song of each type) ≈ 0.2471
- P(playlist has 1 rock, 1 rap, and 1 country song) ≈ 0.2471
- P(playlist has 2 rock songs) ≈ 0.1206
Explanation:
To find the probability of having one song of each type (rock, rap, and country), we need to consider the different ways in which this can happen.
First, let's consider the number of ways David can choose one rock song, one rap song, and one country song from his collection. He can select one rock song from his 6 rock songs in 6 ways. Similarly, he can choose one rap song from his 7 rap songs in 7 ways, and one country song from his 4 country songs in 4 ways.
The total number of ways he can create a playlist with one song of each type is the product of these numbers: 6 x 7 x 4 = 168.
Next, we need to calculate the total number of possible playlists that David can create by choosing any three songs from his collection. This can be found by selecting three songs from his total collection of 6 rock songs, 7 rap songs, and 4 country songs. Using combinations, this can be calculated as 17C3 = 680.
Finally, we can calculate the probability using the formula: Probability = Number of favorable outcomes / Total number of possible outcomes.
P(playlist has one song of each type) = 168 / 680 ≈ 0.2471 (rounded to four decimal places).
2. P(playlist has 1 rock, 1 rap, and 1 country song):
To find the probability of having 1 rock song, 1 rap song, and 1 country song, we can consider the number of ways this can happen.
First, let's calculate the number of ways David can choose one rock song from his 6 rock songs, one rap song from his 7 rap songs, and one country song from his 4 country songs. This can be calculated as 6 x 7 x 4 = 168, which we found in the previous calculation.
The total number of possible playlists, as calculated before, is 680.
Using the formula for probability, we can calculate:
P(playlist has 1 rock, 1 rap, and 1 country song) = 168 / 680 ≈ 0.2471 (rounded to four decimal places).
3. P(playlist has 2 rock songs):
To find the probability of having 2 rock songs in the playlist, we need to consider the number of ways this can happen.
David can choose 2 rock songs from his 6 rock songs in 6C2 = 15 ways. He can then choose 1 song from the remaining 11 songs (7 rap songs and 4 country songs) in 11 ways.
The total number of possible playlists, as calculated before, is 680.
Using the formula for probability, we can calculate:
P(playlist has 2 rock songs) = (6C2 x 11) / 680 ≈ 0.1206 (rounded to four decimal places).
So, the probabilities are approximately:
- P(playlist has one song of each type) ≈ 0.2471
- P(playlist has 1 rock, 1 rap, and 1 country song) ≈ 0.2471
- P(playlist has 2 rock songs) ≈ 0.1206