For a few weeks, a music producer kept track of newly released songs on a music streaming website. He recorded the music genre and number of times the song was played on its release date.The probability that a song was pop is 0.09, the probability that it had 501-1,000 plays is 0.58, and the probability that it was pop or had 501-1,000 plays is 0.64.What is the probability that a randomly chosen song was pop and had 501-1,000 plays?Write your answer as a whole number, decimal, or simplified fraction.
Question
For a few weeks, a music producer kept track of newly released songs on a music streaming website. He recorded the music genre and number of times the song was played on its release date.The probability that a song was pop is 0.09, the probability that it had 501-1,000 plays is 0.58, and the probability that it was pop or had 501-1,000 plays is 0.64.What is the probability that a randomly chosen song was pop and had 501-1,000 plays?Write your answer as a whole number, decimal, or simplified fraction.
Solution
To find the probability that a randomly chosen song was pop and had 501-1,000 plays, we can use the formula for the probability of the union of two events:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Where:
- P(A ∪ B) is the probability of event A or event B occurring (in this case, the song being pop or having 501-1,000 plays)
- P(A) is the probability of event A occurring (the song being pop)
- P(B) is the probability of event B occurring (the song having 501-1,000 plays)
- P(A ∩ B) is the probability of both events A and B occurring (the song being pop and having 501-1,000 plays), which is what we're trying to find.
Substituting the given values into the formula, we get:
0.64 = 0.09 + 0.58 - P(A ∩ B)
Solving for P(A ∩ B), we get:
P(A ∩ B) = 0.09 + 0.58 - 0.64 = 0.03
So, the probability that a randomly chosen song was pop and had 501-1,000 plays is 0.03.
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