Question: NASA in developing the prototypes of a new rocket for launching satellites into space test fires four rockets. There is an equally likely chance that the prototype will fly perfectly (succeed, S) or not fly perfectly (fail, F).

(a) Write down the entire space of possible outcomes of this experiment using S=success, and F=fail.

Let the events, E₁ = exactly three rockets succeed; E₂ = one rocket or more succeed; E₃ = no more than 3 rockets fail; Find the following probabilities:

\(\begin{aligned} & (b)\,\,\,P({{E}_{1}}\cap {{E}_{2}}),\,\,that\,\,is\,\,P({{E}_{1}}\,\,and\,\,{{E}_{2}}) \\ & (c)\,\,\,P({{E}_{1}}\cup {{E}_{2}}),\,\,that\,\,is\,\,P({{E}_{1}}\,or\,\,{{E}_{2}}) \\ & (d)\,\,\,P({{E}_{2}}\cap {{E}_{3}}),\,\,that\,\,is\,\,P({{E}_{2}}\,\,and\,\,{{E}_{3}}) \\ & (e)\,\,\,P({{E}_{2}}\cup {{E}_{3}}),\,\,\,that\,\,is\,\,P({{E}_{2}}\,\,or\,\,{{E}_{3}}) \\ & (f)\,\,\,P({{E}_{1}}\cap {{E}_{3}}),that\,\,is\,\,P({{E}_{1}}\,\,and\,\,{{E}_{3}}) \\ & (g)\,\,\,P({{E}_{1}}\cup {{E}_{3}}),that\,\,is\,\,P({{E}_{1}}\,\,or\,\,{{E}_{3}}) \\ & (h)\,\,\,P({{E}_{1}}\cup {{E}_{2}}\cup {{E}_{3}}),\,\,that\,\,is\,\,P({{E}_{1}}\,\,or\,\,{{E}_{2}}\,\,or\,\,{{E}_{3}}) \\ \end{aligned}\)