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Date May 2021 Marks available 2 Reference code 21M.2.SL.TZ2.9
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number 9 Adapted from N/A

Question

All answers in this question should be given to four significant figures.


In a local weekly lottery, tickets cost $2 each.

In the first week of the lottery, a player will receive $D for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving $10 is 0.03. The grand prize in the first week of the lottery is $1000.

If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be $2000 in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.

Find the value of c.

[2]
a.

Determine whether this lottery is a fair game in the first week. Justify your answer.

[4]
b.

Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of n for the value of the grand prize in the nth week of the lottery.

[2]
c.

The wth week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the wth week, his expected profit is $p.

Find the value of p.

[7]
d.

Markscheme

considering that sum of probabilities is 1             (M1)

0.85+c+0.03+0.002+0.0001=1

0.1179               A1

 

[2 marks]

a.

valid attempt to find ED            (M1)

ED=0×0.85+2×0.1179+10×0.03+50×0.002+1000×0.0001

ED=0.7358            A1

No, not a fair game             A1

for a fair game, ED would be $2 OR players expected winnings are 1.264             R1

 

[4 marks]

b.

recognition of GP with r=2            (M1)

1000×2n-1  OR  5002n           A1

 

[2 marks]

c.

recognizing ED>2            (M1)

correct expression for wth week (or nth week)            (A1)

0×0.85+2×0.1179+10×0.03+50×0.002+1000×2w-1×0.0001

correct inequality (accept equation)            (A1)

0.6358+1000×2w-1×0.0001>2  OR  2n-1>13.642

 

EITHER

n-1>3.76998  OR  w=4.76998            (A1)


OR

ED=1.4358 in week 4  or  ED=2.2358 in week 5            (A1)


THEN

w=5            A1

expected profit per ticket =their ED-2            (M1)

=0.2358            A1

 

[7 marks]

d.

Examiners report

[N/A]
a.
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b.
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c.
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d.

Syllabus sections

Topic 4—Statistics and probability » SL 4.7—Discrete random variables
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Topic 4—Statistics and probability

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