Let \(A_1\), \(A_2\), \(A_3\), \(A_4\), \(A_5\), and \(A_6\) be events in some sample space, each with the same probability \(0<p<1/6\). Define the indicator random variables

\[ I_k = \begin{cases} 0 & \text{if }A_k^c\text{ happens}\\ 1 & \text{if }A_k\text{ happens}, \end{cases} \]

and let \(X\) be their sum

\[ X = I_1 + I_2 + I_3 + I_4+I_5+I_6. \]

1. Assume the events \(A_i\) are mutually disjoint and compute…

   1. \(E(X)\).
   2. \(P(X=1)\).
   3. \(P(X=3)\).

2. Assume the \(A_i\) are independent and compute…

   1. \(E(X)\).
   2. \(P(X=1)\).
   3. \(P(X=3)\).