Solution — 2023 Question 6 Binomial Distribution
(a)
Let represent the random variable with a binomial distribution.
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ParseError: Can't use function '$' in math mode at position 1:
$̲ X \sim \mathrm{B} \left( 8, p \right)
\begin{align*} \mathrm{P}\left( X = 3 \right) &= \binom{8}{3}p^3\left( 1 - p \right)^{8 - 3} \\ &= 56p^3\left( 1 - p \right)^5\; \QED \end{align*}$$ ParseError: {align*} can be used only in display mode.
(b)
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ParseError: Can't use function '$' in math mode at position 1:
$̲ X \sim \mathrm{B} \left( 8, 0.35 \right)
\begin{align*} \mathrm{P}\left(X \geq 4\right) &= 1 - \mathrm{P}\left(X \leq 3\right) \\ &= 1 - 0.706\,40 \\ &= 0.293\,60 \\ &= 0.294\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.
[probability of X \geq 4]
(c)