## Probability Thorem

**Addition Theorem of Probability:**

If A_{1},A_{2}, . . . , A_{k} be k mutually exclusive events on a sample space associated with a random experiment, then the probability of happening one of them is the sum of their individual probabilities. In symbols,

P(A1 A2 . . . Ak) = P(A1) P(A2) . . . P(Ak)

or

P(A1 ∪ A2 ∪. . .∪ Ak) = P(A1) P(A2) . . . P(Ak)

**Proof:** Let total ways = N. Let a_{1},a_{2}, . . . ,ak be the favorable ways to the events A_{1},A_{2}, . . . ,A_{k} respectively. Then the number of favorable ways to any one of the events = a_{1} a_{2} . . . a_{k}

**Conditional Probability:**

The probability that the event B will occur, it being known that A has occurred is called the conditional probability of B and is denoted by P(B/A).In symbols,

Conditional probability of B when A has happened =

**Multiplication Theorem of Probability:
**

**1**. The probability of happening of the two independent events A and B together is equal to the product of their individual probabilities. That is

P( A ∩ B = P(A) P(B )

**2**. If A and B are not independent, then the probability P( A ∩ B ) of their simultaneous occurrence is equal to the product of the probability of A, P (A) and the conditional probability P (B/A). In symbols

P( A ∩ B ) = P (A) P(B/A)

or

P( A ∩ B ) = P(B) P(A/B)

**Specific Formulas:**

**1**. If P( A ∩ B ) = P(A) P(B), then

**2**. If the events A and B are independent

**3**. If the events A and B are independent