II. METEORS AND PAIRWISE INDEPENDENCE Suppose we have $ns students. For each $i \in C, Vidots, n\}$ let $\mathcal(A)_(i]$ be the event that student $i$ gets hit by a meteor. Suppose $P\left\mathcal (A)_{i} ight)=1 / 2^[i+1]$ for all $i \in {1, Vidots, n$ Suppose that the events $\mathcal{A}_{i}$ and $\mathcal A_{j$ are independent whenever $i eq jis a) Compute the exact probability Sp\left\mathcal (A) {1} \cup \mathcal A}_{2} ight]5. b) Compute the exact probability of $P\left[\mathcal (A}_{1 cup \mathcal (A)_ [2] \cup \mathcal A_{3} ight]$ in terms of the (unknown) value $P\left\mathcal{A}_{1) \cap \mathcal{A}_{2} \cap \mathcal A]_(3} ight) .$ (You may need to use the rule $P[A \cup B]=P[A]EP[B]-PIA \cap B]$ several times, or just look up the 'inclusion-exclusion principle.") c) Establish numerical upper and lower bounds $u$ and $L$ such that: $C [1]$ $$ BL Veq P\left[\mathcal(A)_(1} {cup \mathcal{A}_{2} \cup \mathcal A_{3} ight] \leq U1 $$ SP.S0.02