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Date	May 2021	Marks available	7	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]

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

[4]

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 $n th$ week of the lottery.

[2]

The $w th$ 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 $w th$ week, his expected profit is $$ p$ .

Find the value of $p$ .

[7]

Markscheme

considering that sum of probabilities is $1$ (M1)

$0.85 + c + 0.03 + 0.002 + 0.0001 = 1$

$0.1179$ A1

[2 marks]

valid attempt to find $E (D)$ (M1)

$E (D) = (0 \times 0.85) + (2 \times 0.1179) + (10 \times 0.03) + (50 \times 0.002) + (1000 \times 0.0001)$

$E (D) = 0.7358$ A1

No, not a fair game A1

for a fair game, $E (D)$ would be $$ 2$ OR players expected winnings are $1.264$ R1

[4 marks]

recognition of GP with $r = 2$ (M1)

$1000 \times 2^{n - 1}$ OR $500 (2^{n})$ A1

[2 marks]

recognizing $E (D) > 2$ (M1)

correct expression for $w^{th}$ week (or $n^{th}$ week) (A1)

$(0 \times 0.85) + (2 \times 0.1179) + (10 \times 0.03) + (50 \times 0.002) + (1000 \times 2^{w - 1} \times 0.0001)$

correct inequality (accept equation) (A1)

$0.6358 + (1000 \times 2^{w - 1} \times 0.0001) > 2$ OR $2^{n - 1} > 13.642$

EITHER

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

$E (D) = 1.4358$ in week $4$ or $E (D) = 2.2358$ in week $5$ (A1)

THEN

$w = 5$ A1

expected profit per ticket $= their E (D) - 2$ (M1)

$= 0.2358$ A1

[7 marks]

Examiners report

[N/A]

Syllabus sections

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

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