| Date | May 2015 | Marks available | 2 | Reference code | 15M.2.hl.TZ1.3 |
| Level | HL only | Paper | 2 | Time zone | TZ1 |
| Command term | Find | Question number | 3 | Adapted from | N/A |
Question
A mosaic is going to be created by randomly selecting 1000 small tiles, each of which is either black or white. The probability that a tile is white is 0.1. Let the random variable \(W\) be the number of white tiles.
State the distribution of \(W\), including the values of any parameters.
Write down the mean of \(W\).
Find \({\text{P}}(W > 89)\).
Markscheme
\(W \sim B(1000,{\text{ }}0.1)\;\;\;\left( {{\text{accept }}C_k^{1000}{{(0.1)}^k}{{(0.9)}^{1000 - k}}} \right)\) A1A1
Note: First A1 is for recognizing the binomial, second A1 for both parameters if stated explicitly in this part of the question.
[2 marks]
\(\mu ( = 1000 \times 0.1) = 100\) A1
[1 mark]
\({\text{P}}(W > 89) = {\text{P}}(W \ge 90)\;\;\;\left( { = 1 - {\text{P}}(W \le 89)} \right)\) (M1)
\( = 0.867\) A1
Notes: Award M0A0 for \(0.889\)
[2 marks]
Total [5 marks]
Examiners report
Overall this question was well answered. In part (a) a number of candidates did not mention the binomial distribution or failed to state its parameters although they could go on and do the next parts.
In part (b) most candidates could state the expected value.
In part (c) many candidates had problems with inequalities due to the discrete nature of the variable. Most candidates that could deal with the inequality were able to use the GDC to obtain the answer.
