Let $E_1, E_2, E_3$ be events. Let $I_1, I_2, I_3$ be the corresponding indicators such that $I_1 = 1$ if $E_1$ occurs and $I_1 = 0$ otherwise.

(a) Let $I_A = 1 - (1 - I_1)(1 - I_2)$. Verify that $I_A$ is the indicator for the event $A$, where $A = (E_1 \lor E_2)$ (that is, "$E_1$ or $E_2$"), and show that

$$
\operatorname{Pr}(A) = \operatorname{Pr}(E_1) + \operatorname{Pr}(E_2) - \operatorname{Pr}(E_1 \land E_2)
$$

where $(E_1 \land E_2)$ means "$E_1$ and $E_2$".

(b) Find a formula, in terms of $I_1, I_2, I_3$, for $I_B$, the indicator for the event $B$, where $B = (E_1 \lor E_2 \lor E_3)$, and derive a formula for $\operatorname{Pr}(B)$ in terms of $\operatorname{Pr}(E_1), \operatorname{Pr}(E_2), \operatorname{Pr}(E_3), \operatorname{Pr}(E_1 \land E_2), \operatorname{Pr}(E_1 \land E_3), \operatorname{Pr}(E_2 \land E_3), \operatorname{Pr}(E_1 \land E_2 \land E_3)$.