Independent Events

Two probabilistic events $H_1$ and $H_2$ are said to be independent if the probability of $H_1$ and $H_2$ occurring together equals the product of the probabilities of $H_1$ and $H_2$ individually, i.e.,

$$p(H_1 H_2) = p(H_1)(H_2)$$

where $p(H_1 H_2)$ denotes the probability of $H_1$ and $H_2$ occurring together.

Example: Successive coin tosses are normally independent. Therefore, the probability of getting heads twice in a row is given by

$$p(H H) = p(H)p(H) = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}.$$