Conditional Probability or Bayes Theorem

Question

I have this question and it has to do with either Bayes' Theorem or Conditional Probability. Any help in solving it?

There is a 65% chance of John passing mathematics. There is a 35% chance that John will pass both mathematics and statistics. There is a 70% chance that he will pass either mathematics or statistics or both. John has been informed that he has passed statistics, what is the probability that he will pass mathematics?

Answer 1

Let $M$ and $S$ be the events that John passes his math and statistics exams respectively.

Let $M^C$ and $S^C$ be the events that John fails his math and statistics exams respectively. From the given information, we have the following table.

|       |   S | S^C | Total |
|-------|-----|-----|-------|
|     M | 35% |     |   65% |
|   M^C |     | 30% |       |
|-------|-----|-----|-------|
| Total |     |     |  100% |

Complete this table.

|       |   S | S^C | Total |
|-------|-----|-----|-------|
|     M | 35% | 30% |   65% |
|   M^C |  5% | 30% |   35% |
|-------|-----|-----|-------|
| Total | 40% | 60% |  100% |

$\text{Required probability} = P(M|S)=\frac{P(MS)}{S}=\frac{0.35}{0.4}=\frac78$