| Date | November 2012 | Marks available | 2 | Reference code | 12N.2.hl.TZ0.7 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 7 | Adapted from | N/A |
Question
Kathy plays a computer game in which she has to find the path through a maze within a certain time. The first time she attempts the game, the probability of success is known to be 0.75. In subsequent attempts, if Kathy is successful, the difficulty increases and the probability of success is half the probability of success on the previous attempt. However, if she is unsuccessful, the probability of success remains the same. Kathy plays the game three times consecutively.
Find the probability that she is successful in all three games.
Assuming that she is successful in the first game, find the probability that she is successful in exactly two games.
Markscheme
\({\text{P(WWW)}} = 0.75 \times 0.375 \times 0.1875 = 0.0527{\text{ (3sf) }}\left( {\frac{3}{4} \times \frac{3}{8} \times \frac{3}{{16}} = \frac{{27}}{{512}}} \right)\) (M1)A1
[2 marks]
(M1)(A1)
Note: Award M1 for any reasonable attempt to use a tree diagram showing that three games were played (do not award M1 for tree diagrams that only show the first two games) and A1 for the highlighted probabilities.
\(\left. {{\text{P(wins 2 games}}\,} \right|{\text{wins first game)}} = \frac{{{\text{P(WWL, WLW)}}}}{{{\text{P(wins first game)}}}}\) (M1)
\( = \frac{{0.75 \times 0.375 \times 0.8125 + 0.75 \times 0.625 \times 0.375}}{{0.75}}\) (A1)(A1)
\( = 0.539{\text{ (3sf) }}\left( {{\text{or }}\frac{{69}}{{128}}} \right)\) A1
Note: Candidates may use the tree diagram to obtain the answer without using the conditional probability formula, ie,
\(\left. {{\text{P(wins 2 games}}\,} \right|{\text{wins first game)}} = 0.375 \times 0.8125 + 0.625 \times 0.375 = 0.539.\)
[6 marks]
Examiners report
Part (a) was generally successful to most candidates; however the conditional probability was proved difficult to many candidates either because the unconditional probability of two correct games was found or the success in the second and third game was included. Many candidates used a clear tree diagram to calculate the corresponding probabilities. However other candidates frequently tried to do the problem without drawing a tree diagram and often had incorrect probabilities. It was sad to read many answers with probabilities greater than 1.
Part (a) was generally successful to most candidates; however the conditional probability was proved difficult to many candidates either because the unconditional probability of two correct games was found or the success in the second and third game was included. Many candidates used a clear tree diagram to calculate the corresponding probabilities. However other candidates frequently tried to do the problem without drawing a tree diagram and often had incorrect probabilities. It was sad to read many answers with probabilities greater than 1.
