**Bayes’ Theorem of Probability, Conditional Probability and Intersection of Probability.**

**Conditional Probability and Bayes’ Theorem**

The conditional probability is written as P(A|B), which means, the probability of A from given the event of Probability B has already occurred. It can be easily explained from the below Venn diagram.

So, from the above equation we can derive

P(A & B) = P(A|B) * P(B) ……………1

Similarly

P(B & A) = P(B|A)*P(A) …………….2

Since P(A&B) and P(B&A) are same. So equations 1 and 2 are same. That is

P(B|A)*P(A) = P(A|B) * P(B)

Hence

P(B|A) ={ P(A|B) * P(B)} / P(A) …………… Bayes’ theorem.

The Bayes’ theorem has application in nearly all branches of science. Given the historical data we can predict about future. It is used in medical science, engineering and even in courts of justice.

**Intersection of Probability**

The intersection of probability can be understood from below table.

Toothache | Toothache | |

Cavity | 0.04 | 0.06 |

Cavity | 0.01 | 0.89 |

P(cavity & toothache) = 0.04

P(cavity & Toothache) = 0.06

P(cavity) = P(cavity & toothache) + P(cavity & Toothache) ……………3

= 0.04 + 0.06

= 0.1

From equation 3, we conclude that if we have two factors A and B then

P(A) = P(A & B) + P(A & B)

Moreover, by using equation 1, we can write

P(A) = P(A|B) * P(B) + P(B|A)*P(A)