VCE Methods Probability Application Task 6

Question 1 [2022 Exam 2 Section B Q3]

Mika is flipping a coin. The unbiased coin has a probability of \( \frac{1}{2} \) of landing on heads and \( \frac{1}{2} \) 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.

a. i. Find \( \Pr(X = 5) \). 1 mark

ii. Find \( \Pr(X \geq 2) \). 1 mark

iii. Find \( \Pr(X \geq 2 \mid X < 5) \), correct to three decimal places. 2 marks

iv. Find the expected value and the standard deviation for \( X \). 2 marks

The height reached by each of Mika’s coin flips is given by a continuous random variable, \( H \), with the probability density function

\[ f(h) = \begin{cases} ah^2 + bh + c, & 1.5 \leq h \leq 3 \\ 0, & \text{otherwise} \end{cases} \]

where \( h \) is the vertical height reached by the coin flip, in metres, between the coin and the floor, and \( a, b, c \in \mathbb{R} \).

b. i. State the value of the definite integral \( \int_{1.5}^{3} f(h) \, dh \). 1 mark

ii. Given that \( \Pr(H \leq 2) = 0.35 \) and \( \Pr(H \geq 2.5) = 0.25 \), find the values of \( a \), \( b \), and \( c \). 3 marks

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

c. Mika’s sister Bella also has a coin. On each flip, Bella’s coin has a probability of \( p \) of landing on heads and \( (1 - p) \) 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.

i. Is the random variable \( \hat{P} \) discrete or continuous? Justify your answer. 1 mark

ii. If \( \hat{p} = 0.4 \), find an approximate 95% confidence interval for \( p \), correct to three decimal places. 1 mark

iii. 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