| Date | November 2009 | Marks available | 20 | Reference code | 09N.2.hl.TZ0.13 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Calculate and Determine | Question number | 13 | Adapted from | N/A |
Question
In each round of two different games Ying tosses three fair coins and Mario tosses two fair coins.
(a) The first game consists of one round. If Ying obtains more heads than Mario, she receives $5 from Mario. If Mario obtains more heads than Ying, he receives $10 from Ying. If they obtain the same number of heads, then Mario receives $2 from Ying. Determine Ying’s expected winnings.
(b) They now play the second game, where the winner will be the player who obtains the larger number of heads in a round. If they obtain the same number of heads, they play another round until there is a winner. Calculate the probability that Ying wins the game.
Markscheme
(a) Ying:
(M1)A1
Mario:
(M1)A1
\({\text{P(Ying wins)}} = \frac{1}{8} + \frac{3}{8}\left( {\frac{2}{4} + \frac{1}{4}} \right) + \frac{3}{8} \times \frac{1}{4}\)
\( = \frac{{16}}{{32}}\) (M1)A1
\({\text{P(Mario wins)}} = \frac{1}{4}\left( {\frac{3}{8} + \frac{1}{8}} \right) + \frac{2}{4} \times \frac{1}{8}\)
\( = \frac{6}{{32}}\) (M1)A1
\({\text{P(draw)}} = 1 - \frac{{16}}{{32}} - \frac{6}{{32}}\)
\( = \frac{{10}}{{32}}\) A1
Ying’s winnings:
expected winnings \( = 5\left( {\frac{{16}}{{32}}} \right) - 105\left( {\frac{6}{{32}}} \right) - 25\left( {\frac{{10}}{{32}}} \right)\) M1A1
\( = 0\) A1
[12 marks]
(b) \({\text{P(Ying wins on 1st round)}} = \frac{1}{2}\) (A1)
\({\text{P(Ying wins on 2st round)}} = \frac{5}{{16}} \times \frac{1}{2}\) (M1)(A1)
\({\text{P(Ying wins on 3rd round)}} = {\left( {\frac{5}{{16}}} \right)^2} \times \frac{1}{2}\) etc. (A1)
\({\text{P(Ying wins)}} = \frac{1}{2} + \frac{5}{{16}} \times \frac{1}{2} + {\left( {\frac{5}{{16}}} \right)^2} \times \frac{1}{2} + ...\) (M1)
\( = \frac{{\frac{1}{2}}}{{1 - \frac{5}{{16}}}}\) M1A1
\( = \frac{8}{{11}}\) ( \( =0.727\)) A1
[8 marks]
Total [20 marks]
Examiners report
There were some good attempts at this question, but there were also many candidates that were unable to maintain a clearly presented solution and consequently were unable to obtain marks that they should have been able to secure. Those that attempted part (b) usually made a good attempt.
