Mika is flipping a coin. The unbiased coin has a probability of \(\dfrac{1}{2}\) of landing on heads and \(\dfrac{1}{2}\) of landing on tails.
Let \(X\) be the binomial random variable representing the number of times that the coin lands on heads.
Mika flips the coin five times.
 i. Find \(\text{Pr}(X=5)\). (1 mark)
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ii. Find \(\text{Pr}(X \geq 2).\) (1 mark)
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 iii. Find \(\text{Pr}(X \geq 2  X<5)\), correct to three decimal places. (2 marks)
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 iv. Find the expected value and the standard deviation for \(X\). (2 marks)
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The height reached by each of Mika's coin flips is given by a continuous random variable, \(H\), with the probability density function
\(f(h)=\begin{cases} ah^2+bh+c &\ \ 1.5\leq h\leq 3 \\ \\ 0 &\ \ \text{elsewhere} \\ \end{cases}\)
where \(h\) is the vertical height reached by the coin flip, in metres, between the coin and the floor, and \(a, b\) and \(c\) are real constants.
 i. State the value of the definite integral \(\displaystyle\int_{1.5}^3 f(h)\,dh\). (1 mark)
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 ii. Given that \(\text{Pr}(H \leq 2)=0.35\) and \(\text{Pr}(H \geq 2.5)=0.25\), find the values of \(a, b\) and \(c\). (3 marks)
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 iii. The ceiling of Mika's room is 3 m above the floor. The minimum distance between the coin and the ceiling is a continuous random variable, \(D\), with probability density function \(g\).
 The function \(g\) is a transformation of the function \(f\) given by \(g(d)=f(rd+s)\), where \(d\) is the minimum distance between the coin and the ceiling, and \(r\) and \(s\) are real constants.
 Find the values of \(r\) and \(s\). (1 mark)
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 Mika's sister Bella also has a coin. On each flip, Bella's coin has a probability of \(p\) of landing on heads and \((1p)\) of landing on tails, where \(p\) is a constant value between 0 and 1 .
 Bella flips her coin 25 times in order to estimate \(p\).
 Let \(\hat{P}\) be the random variable representing the proportion of times that Bella's coin lands on heads in her sample.

 Is the random variable \(\hat{P}\) discrete or continuous? Justify your answer. (1 mark)
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 If \(\hat{p}=0.4\), find an approximate 95% confidence interval for \(p\), correct to three decimal places. (1 mark)
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 Bella knows that she can decrease the width of a 95% confidence interval by using a larger sample of coin flips.
 If \(\hat{p}=0.4\), how many coin flips would be required to halve the width of the confidence interval found in part c.ii.? (1 mark)
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 Is the random variable \(\hat{P}\) discrete or continuous? Justify your answer. (1 mark)