Question

Are Independent Events Transitive

Answer 1

Flip two coins. $\alpha$ is first coin heads, $\beta$ is second coin heads and $\gamma$ is that the coins agree. Each of these events are independent of the other two and has probability $\frac{1}{2}$. $\alpha \cap \beta$ implies that both coins are heads, so $\gamma$ is true event and hence not independent of $\alpha \cap \beta$ . Also $\alpha \cup \beta$ implies that there is at least one head, hence two tails can't happen and the probability of $\gamma$ is just $\frac{1}{3}$, therefore $\gamma$ is not independent of $\alpha \cup \beta$.

Hence, Independent events are not transitive.