Solution — 2017 Question 7 Binomial Distribution
Let represent the number of cartridges that last one week or more out of
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ParseError: Can't use function '$' in math mode at position 1:
$̲ X \sim \mathrm{B} \left( 8, 0.7 \right)
\begin{align*} \mathrm{P}\left(X = 5\right) &= 0.254\,12 \\ &= 0.254\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.
[probability of X = 5]
Let represent the number of cartridges that last less than one week out of
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ParseError: Can't use function '$' in math mode at position 1:
$̲ Y \sim \mathrm{B} \left( 8, 0.300,00 \right)
\begin{align*} \mathrm{P}\left(Y \geq 4\right) &= 1 - \mathrm{P}\left(Y \leq 3\right) \\ &= 1 - 0.805\,90 \\ &= 0.194\,10 \\ &= 0.194\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.
[probability of Y \geq 4]
Let represent the number of boxes where at least half of the cartridges last less than one week, out of boxes.
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ParseError: Can't use function '$' in math mode at position 1:
$̲ B \sim \mathrm{B} \left( 6, 0.194,10 \right)
\begin{align*} \mathrm{P}\left(B \leq 2\right) &= 0.908\,23 \\ &= 0.908\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.
[probability of B \leq 2]