## 8.10 Conditional Probability

### 8.10.1 Conditional Formula

**Bayes Theorem**

\(P(B|A)=\frac{P(A \cap B)}{P(A)}\)

### 8.10.2 Conditional Problem # 01

Probability that it rains today is 40%, that tomorrow is 50%, and that on both days is 30%. If it rains today, what is the probability that it would rain tomorrow?

- \(P (T) = 0.4, P(M) = 0.5, P(T\cap M)=0.3\)
- \(P(M|T)=?\)
- \(P(M|T)=\frac{P(T\cap M)}{P(T)}\)
- \(\frac{0.3}{0.4}\)

### 8.10.3 Conditional Problem # 02

In a college, there are 100 students, of whom 30 play football, 40 play cricket, and 20 play both. A student is selected randomly. If he plays cricket, what is the probability that he plays football?

\(P(F)=0.3\)

\(P(C)=0.4\)

\(P(F \cap C)=0.2\)

\(P(F|C)=?\)

- \(P(F|C)=\frac{P(F \cap C)}{P(C)}=\) 0.5

### 8.10.4 Conditional Problem # 03

\(S=\) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

If a number is picked randomly and known it an even number, What is the probability that it is more than 6?

### 8.10.5 Conditional Problem # 04

In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. The city has a facial recognition software. If the camera scans a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time. If the camera scans a non-terrorist, a bell will not ring 99% of the time, but it will ring 1% of the time.

If the bell rings, what is the probability that a terrorist is caught?

About 99 of the 100 terrorists will trigger the alarmâ€”and so will about 9,999 of the 999,900 non-terrorists. Therefore, about 10,098 people will trigger the alarm, among which about 99 will be terrorists. So, the probability that a person triggering the alarm actually is a terrorist, is only about 99 in 10,098, which is less than 1%