S1 January 2013 Q6
6. A fair blue die has faces numbered 1, 1, 3, 3, 5 and 5. The random variable \(B\) represents the score when the blue die is rolled.
A second die is red and the random variable \(R\) represents the score when the red die is rolled.
The probability distribution of \(R\) is
| \(r\) | 2 | 4 | 6 |
|---|---|---|---|
| \(\mathrm{P}(R = r)\) | \(\dfrac{2}{3}\) | \(\dfrac{1}{6}\) | \(\dfrac{1}{6}\) |
Tom invites Avisha to play a game with these dice.
Tom spins a fair coin with one side labelled 2 and the other side labelled 5. When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is greater than the number showing on the coin, Avisha wins, otherwise Tom wins.
Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.
| Scheme | Marks | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| B1 B1 | ||||||||
| (2) |
Notes
1st B1 for correctly identifying values of \(b\) as 1, 3, 5 or 1,1,3,3,5,5
2nd B1 for probabilities all \(= \frac{1}{3}\) or exact equivalent (or of course 6 cases of \(\frac{1}{6}\))
Any correct probability distribution or probability function is 2/2. Must be in part (a)
| Scheme | Marks |
|---|---|
| Discrete Uniform {distribution} | B1 |
| (1) |
Notes
B1 for "Discrete Uniform". Both words required.
| Scheme | Marks |
|---|---|
| \([\mathrm{E}(B) =]\ 3\) (by symmetry) | B1 |
| (1) |
Notes
B1 for answer of 3 o.e. Accept \(\mathrm{E}(X) = 3\)
| Scheme | Marks |
|---|---|
| \([\mathrm{E}(R) =]\ 2\times\dfrac{2}{3} + 4\times\dfrac{1}{6} + 6\times\dfrac{1}{6}\) | M1 |
| \(= \underline{\mathbf{3}}\) | A1 |
| (2) |
Notes
M1 for an attempt at correct formula. At least 2 correct products seen. If later divide by \(n\ (\ne 1)\) M0
A1 for an answer of 3. Correct answer only scores both marks.
| Scheme | Marks |
|---|---|
| \([\mathrm{E}(R^2) =]\ 2^2\times\dfrac{2}{3} + 4^2\times\dfrac{1}{6} + 6^2\times\dfrac{1}{6} \quad \left[= \dfrac{34}{3}\right]\) | M1 |
| \([\mathrm{Var}(R) =]\ \dfrac{34}{3} - 3^2 = \dfrac{7}{3}\) (or any exact equivalent. NB 2.33 is A0) | dM1, A1 |
| (3) |
Notes
1st M1 for a correct attempt at \(\mathrm{E}(R^2)\). At least 2 correct products seen. Condone \(\mathrm{Var}(R) =\) etc. May be implied by sight of \(\frac{34}{3}\) or 11.3 or better.
2nd dM1 Dep. on 1st M1 for clear attempt at \(\mathrm{E}(R^2) - [\mathrm{E}(R)]^2\). Must see their values used.
NB \(\mathrm{Var}(R) = \mathrm{E}(R^2) - [\mathrm{E}(R)]^2 = \text{"}\tfrac{34}{3}\text{"} - \text{"}3\text{"}\) is M1M0A0 since do not use their \([\mathrm{E}(R)]^2\)
| Scheme | Marks |
|---|---|
| Coin lands on 2, choose blue die; coin lands on 5 choose red die | B2/1/0 |
| P(Avisha wins) \(= \dfrac{1}{2}\times\left(\dfrac{1}{3} + \dfrac{1}{3}\right) + \dfrac{1}{2}\times\dfrac{1}{6}\) | M1 |
| \(= \dfrac{5}{12}\) (allow awrt 0.417) | A1 |
| (4) | |
| (13 marks) |
Notes
B2/1/0 Both correct B1B1, one correct B1B0. Do not use B0B1[e.g. always red or RR is B1B0]
NB Allow other descriptions of the die e.g. 1st or fair for blue, 2nd for red if they are clear.
M1 for evaluating correct probabilities i.e. only \(\frac{1}{3}, \frac{1}{12}\) seen or if incorrect choice made:
M1 for an answer of : if choose RR (\(\frac{1}{4}\)), if choose BB (\(\frac{1}{3}\)), if choose RB (\(\frac{1}{6}\))
NB \(\frac{5}{12}\) as answer scores M1A1. Need to see choices of die stated for B marks.