A commercial space low earth orbit satellite company employs an external firm of engineers to audit their designs before they attempt to put them in orbit. In the past 90% of their designs have proved to be successful whereas 10% have proven unsuccessful. The firm of engineers has given positive reviews to 95% of the successful designs and positive reviews to 15% of the unsuccessful designs. Given a positive review from the firm of engineers what is the probability that a design will be successful (round your answer to four decimal places)?
Question
A commercial space low earth orbit satellite company employs an external firm of engineers to audit their designs before they attempt to put them in orbit. In the past 90% of their designs have proved to be successful whereas 10% have proven unsuccessful. The firm of engineers has given positive reviews to 95% of the successful designs and positive reviews to 15% of the unsuccessful designs. Given a positive review from the firm of engineers what is the probability that a design 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 design is successful
- U as the event that a design is unsuccessful
- P as the event that a design gets a positive review
We know from the problem:
- P(S) = 0.90 (probability of a design being successful)
- P(U) = 0.10 (probability of a design being unsuccessful)
- P(P|S) = 0.95 (probability of a positive review given a successful design)
- P(P|U) = 0.15 (probability of a positive review given an unsuccessful design)
We want to find P(S|P), the probability of a design being successful given a positive review.
According to Bayes' theorem:
P(S|P) = P(P|S) * P(S) / P(P)
We also know that P(P) can be calculated as:
P(P) = P(P and S) + P(P and U) = P(P|S) * P(S) + P(P|U) * P(U)
Substituting the known values:
P(P) = 0.95 * 0.90 + 0.15 * 0.10 = 0.855 + 0.015 = 0.87
Now we can find P(S|P):
P(S|P) = P(P|S) * P(S) / P(P) = 0.95 * 0.90 / 0.87 = 0.9804
So, given a positive review from the firm of engineers, the probability that a design will be successful is approximately 0.9804 or 98.04%.
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