Probability - Bayes Theorem

Question

1. Rodney's Repair Service has a lug nut tightening machine that works well 85% of the time. They got a new machine that works well 95% of the time. Each machine is used 50% of the time. Use Bayes' Theorem to find the probability. If necessary, round your calculations to the nearest thousandths, and write your answer as a percent.

The new machine malfunctioned.

The new machine will malfunction __% of the time.

2. Rodney's Repair Service has a lug nut tightening machine that works well 88% of the time. They got a new machine that works well 99% of the time. Each machine is used 50% of the time. Use Bayes' Theorem to find the probability. If necessary, round your calculations to the nearest thousandths, and write your answer as a percent.

The old machine malfunctioned.

The old machine will malfunction __% of the time.

3. Rodney's Repair Service has a lug nut tightening machine that works well 89% of the time. They got a new machine that works well 96% of the time. Each machine is used 50% of the time. Use Bayes' Theorem to find the probability. If necessary, round your calculations to the nearest thousandths, and write your answer as a percent.

The old machine worked well.

The old machine will work well __% of the time.

4. Rodney's Repair Service has a lug nut tightening machine that works well 89% of the time. They got a new machine that works well 96% of the time. Each machine is used 50% of the time. Use Bayes' Theorem to find the probability. If necessary, round your calculations to the nearest thousandths, and write your answer as a percent.

The new machine worked well.

The new machine will work well __% of the time.

I know that I need to use this formula: however I am not sure how.

Answer 1

Perhaps they are asking you to calculate the probability that the machine was the new machine given that the machine malfunctioned. Let $M$ be malfunction, $O$ be old machine used, $N$ be new machine used. Then

$$P(N|M)=\frac{P(M|N)P(N)}{P(M|N)P(N)+P(M|O)P(O)}=\frac{.05(.5)}{.05(.5)+.15(.5)}=.25$$

So if the machine malfunctioned, there's a $\frac 14$ chance that it was the new machine. Now the rest of the problems are similar.