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.,

$$\hat{p}(H_1 H_2) = \hat{p}(H_1)(H_2)$$ (C.2)

where $\hat{p}(H_1 H_2)$ 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

$$\hat{p}(H H) = \hat{p}(H)\hat{p}(H) = \frac{1}{2}\cdot\frac{1}{2} = \frac{1}{4}.$$ (C.3)