You are a DJ at a wedding. A playlist contains 10 pop songs and 20 country songs. You set the playlist to select songs at random. Once a song is played, the same song will not play again. Find the probability that the first two songs to play are both country songs. Express your first answer as a fraction in simplest form, and round your percent answer to the nearest tenth. The probability is , or about %.
Question
You are a DJ at a wedding. A playlist contains 10 pop songs and 20 country songs. You set the playlist to select songs at random. Once a song is played, the same song will not play again. Find the probability that the first two songs to play are both country songs. Express your first answer as a fraction in simplest form, and round your percent answer to the nearest tenth. The probability is , or about %.
Solution 1
The total number of songs in the playlist is 10 (pop songs) + 20 (country songs) = 30 songs.
The probability that the first song is a country song is 20 (country songs) / 30 (total songs) = 2/3.
After the first country song is played, there are now 19 country songs left and a total of 29 songs left in the playlist.
So, the probability that the second song is also a country song is 19 (remaining country songs) / 29 (remaining total songs) = 19/29.
The probability that both the first and second songs are country songs is the product of the two individual probabilities, so (2/3) * (19/29) = 38/87.
As a percentage, this is approximately 43.7%, rounded to the nearest tenth.
So, the probability that the first two songs to play are both country songs is 38/87, or about 43.7%.
Solution 2
To find the probability that the first two songs to play are both country songs, we need to use the formula for probability which is:
P(A) = Number of favorable outcomes / Total number of outcomes
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The total number of songs in the playlist is 10 (pop songs) + 20 (country songs) = 30 songs.
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The probability that the first song is a country song is 20 (country songs) / 30 (total songs) = 2/3.
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After the first country song is played, there are now 19 country songs and 29 total songs left. So, the probability that the second song is a country song is 19/29.
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The probability that both events (first and second songs are country songs) occur is found by multiplying the probabilities of each event. So, the probability that the first two songs are country songs is (2/3) * (19/29) = 38/87.
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To express this as a percentage, we divide 38 by 87 and multiply by 100, which gives us approximately 43.7%.
So, the probability that the first two songs to play are both country songs is 38/87, or about 43.7%.
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