A streaming TV service classifies its science fiction programs before they go to air. In the past it was discovered that 30% of shows was successful and 70% unsuccessful. Seventy percent of successful shows had positive feedback from focus groups and 40% of unsuccessful shows had positive feedback. Given that a show has positive feedback from the focus group what is the probability that it will be successful (round your answer to four decimal places)?
Question
A streaming TV service classifies its science fiction programs before they go to air. In the past it was discovered that 30% of shows was successful and 70% unsuccessful. Seventy percent of successful shows had positive feedback from focus groups and 40% of unsuccessful shows had positive feedback. Given that a show has positive feedback from the focus group what is the probability that it will be successful (round your answer to four decimal places)?
Solution
This is a problem of conditional probability and can be solved using Bayes' theorem.
Let's denote:
- S as the event that a show is successful
- F as the event that a show has positive feedback
We are given the following probabilities:
- P(S) = 0.30, the probability that a show is successful
- P(S') = 0.70, the probability that a show is not successful
- P(F|S) = 0.70, the probability that a successful show has positive feedback
- P(F|S') = 0.40, the probability that an unsuccessful show has positive feedback
We want to find P(S|F), the probability that a show is successful given that it has positive feedback.
According to Bayes' theorem:
P(S|F) = P(F|S) * P(S) / P(F)
We don't know P(F), the probability that a show has positive feedback, but we can calculate it using the law of total probability:
P(F) = P(F and S) + P(F and S') = P(F|S) * P(S) + P(F|S') * P(S') = 0.70 * 0.30 + 0.40 * 0.70 = 0.21 + 0.28 = 0.49
Now we can substitute P(F) into Bayes' theorem:
P(S|F) = P(F|S) * P(S) / P(F) = 0.70 * 0.30 / 0.49 = 0.4286
So, given that a show has positive feedback from the focus group, the probability that it will be successful is approximately 0.4286, or 42.86%.
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