15 In this question you must show detailed reasoning.
The random variable \(X\) has probability distribution defined as follows.
\(\mathrm { P } ( X = x ) = \begin{cases} \frac { 15 } { 64 } \times \frac { 2 ^ { x } } { x ! } & x = 2,3,4,5 ,
0 & \text { otherwise. } \end{cases}\)
- Show that \(\mathrm { P } ( X = 2 ) = \frac { 15 } { 32 }\).
The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
- Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 9\), determine the probability that at least one of these three values is equal to 2 .
Freda chooses values of \(X\) at random until she has obtained \(X = 2\) exactly three times. She then stops.
- Determine the probability that she chooses exactly 10 values of \(X\).
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