Download Credits: Four

90643 Solve straightforward problems involving probability
Credits: Four
Answer ALL questions in the spaces provided in this booklet.
You should show ALL working.
Check that this booklet has pages 2–5 in the correct order and that none of these pages is
blank.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE
EXAMINATION.
For Assessor’s use only
Achievement Criteria
Achievement
Solve straight forward
problems involving
probability.
Achievement with Merit
Solve probability problems.
Overall level of Performance
Achievement with Excellence
Apply probability theory.
Page 2 of 6
You are advised to spend 40 minutes answering the questions in this booklet.
Show ALL working
Question One
For the launch of the new zPod, a music store is running a competition.
Each customer will choose four songs of the 10 different songs available on a special zPod.
The zPod will then shuffle the songs and play them in a random order.
If the first four songs played are the same as the ones the customer has chosen, they will win a discount
of $1000.
If any three of the first four songs played are the same as the ones the customer has chosen, they will
win a discount of $100.
If any two of the first four songs played are the same as the ones the customer has chosen, they will win
a discount of $50.
For anything else, they will win no discount.
The table below shows the probability distribution of the random variable D, the amount of discount
won.
d($)
P(D = d)
(a)
0
95
210
50
90
210
100
24
210
1000
1
210
Suppose they have 520 customers over the weekend that the competition runs.
Calculate the total amount of money that the music store would expect to discount over the
weekend.
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(b)
A husband and wife each have a turn at the competition.
Calculate the probability that at least one of them wins a discount of at least $100.
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(c)
The manager of the music store has now decided to give a discount of $20 if any one of the first
four songs played is the same as the ones the customer has chosen.
Two friends are going to participate in the competition.
Calculate the probability that they both win a $20 discount.
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(d)
The manager now wants to change the amount of the discounts won in the competition.
The probability distribution of the random variable D, the new amount of discount won, is shown
in the table below:
d($)
P(D = d)
0
95
210
k
90
210
2k
24
210
10k
1
210
Find the value of k so that the variance of the amount of discount won is $100.
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Page 5 of 6
Question Two
Over one weekend, a cell phone store had a total of 520 customers:
 285 of the customers purchased at least one pre-paid card
 92 of the customers purchased at least one cell phone
 302 of the customers purchased at least one cell phone or at least one pre-paid card
 The rest of the customers did not purchase any cell phones or pre-paid cards
(a)
Calculate the probability that a randomly selected customer from that weekend purchased at least
one cell phone and at least one pre-paid card.
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(b)
Are the events “a customer purchased at least one cell phone” and “a customer purchased at least
one pre-paid card” independent? Justify your answer.
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(c)
The owner of the cell phone store claims that customers are twice as likely to purchase at least
one pre-paid card if they purchase at least one cell phone than if they don’t purchase a cell
phone.
Based on the data provided, is this claim justified? You must support your answer with statistical
reasoning.
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