Question 14 Marks
A fair coin is tossed 8 times. Find the probability that :
(i) it shows no head.
(ii) it shows head at least once.
(i) it shows no head.
(ii) it shows head at least once.
Answer
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2 questions · self-marked practice — reveal the answer and mark yourself.
| $X =x$ | 0 | 1 | 2 | 3 | 4 |
| $P (x = x)$ | 0.08 | 0.15 | 0.45 | 0.27 | 0.05 |
$E(X)=\sum x_i \cdot P\left(x_i\right)$
$\begin{aligned} & =0(0.08)+1(0.15)+2(0.45)+3(0.27)+4(0.05) \\ & =0+0.15+0.9+0.81+0.2=2.06\end{aligned}$
$E\left(X^2\right)=\sum x_i^2 \cdot P\left(x_i\right)$
$\begin{aligned} & =0(0.08)+1^2(0.15)+2^2(0.45)+3^2(0.27)+4^2(0.05) \\ & =0+0.15+1.8+2.43+0.8=5.18\end{aligned}$
$\begin{aligned} & \operatorname{Var}(X)=E\left(X^2\right)-[E(X)]^2 \\ & =5.18-(2.06)^2 \\ & =5.18-4.2436 \\ & =0.9364\end{aligned}$