| Date | November 2014 | Marks available | 4 | Reference code | 14N.2.hl.TZ0.12 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 12 | Adapted from | N/A |
Question
Ava and Barry play a game with a bag containing one green marble and two red marbles. Each player in turn randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.
Find the probability that Ava wins on her first turn.
Find the probability that Barry wins on his first turn.
Find the probability that Ava wins in one of her first three turns.
Find the probability that Ava eventually wins.
Markscheme
\({\text{P(Ava wins on her first turn)}} = \frac{1}{3}\) A1
[1 mark]
\({\text{P(Barry wins on his first turn)}} = {\left( {\frac{2}{3}} \right)^2}\) (M1)
\( = \frac{4}{9}\;\;\;( = 0.444)\) A1
[2 marks]
\(P\)(Ava wins in one of her first three turns)
\( = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3}\) M1A1A1
Note: Award M1 for adding probabilities, award A1 for a correct second term and award A1 for a correct third term.
Accept a correctly labelled tree diagram, awarding marks as above.
\( = \frac{{103}}{{243}}\;\;\;( = 0.424)\) A1
[4 marks]
\({\text{P(Ava eventually wins)}} = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \ldots \) (A1)
using \({S_\infty } = \frac{a}{{1 - r}}\) with \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\) (M1)(A1)
Note: Award (M1) for using \({S_\infty } = \frac{a}{{1 - r}}\) and award (A1) for \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\).
\( = \frac{3}{7}\;\;\;( = 0.429)\) A1
[4 marks]
Total [11 marks]
Examiners report
Parts (a) and (b) were straightforward and were well done.
Parts (a) and (b) were straightforward and were well done.
Parts (c) and (d) were also reasonably well done.
Parts (c) and (d) were also reasonably well done. A pleasingly large number of candidates recognized that an infinite geometric series was required in part (d).
