Mathematical Methods — Question 15

QCAA 2023, Paper 1 · 4 marks

Q15 · 2023 · Technology-freeSimple familiar4 marks

QUESTION 15 (4 marks)

In a certain game, players throw one water balloon at a target. There is a one in four chance of hitting the target.

State the probabilities of all the possible outcomes for one throw at the target.

[2 marks]

Let

H

be the discrete random variable for one of the possible outcomes. Determine the mean and variance of the distribution of random variable

H

when 20 players throw a water balloon at the target.

[2 marks]

WORKED SOLUTION

4 marks

ANSWER

P(\text{hit})=\frac14

P(\text{miss})=\frac34

; b) mean

=5

, variance

=\frac{15}{4}=3\frac34

Worked solution

a) There are two outcomes: hit or miss.

P(\text{hit})=\frac14,\qquad P(\text{miss})=\frac34.

b) Using

H

to represent the number of hits, the mean of a binomial distribution is

np

np=20\times\frac14=5.

The variance is

np(1-p)

np(1-p)=5\times\frac34=\frac{15}{4}=3\frac34.

correctly states the probability of a ‘hit’

[1 mark]

correctly states the probability of a ‘miss’

[1 mark]

determines the mean

[1 mark]

determines the variance

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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