AS June 2018 Q3
3. A fair six-sided black die has faces numbered 1, 2, 2, 3, 3 and 4
The random variable \(B\) represents the score when the black die is rolled.
A white die has 6 faces numbered 1, 1, 2, 4, 5 and \(c\) where \(c \gt 5\)
The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by
| \(w\) | 1 | 2 | 4 | 5 | \(c\) |
|---|---|---|---|---|---|
| \(\mathrm{P}(W = w)\) | \(a + b\) | \(a\) | 0.3 | \(a\) | \(b\) |
Greg and Nilaya play a game with these dice.
Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
The probability of Greg winning the game is \(\dfrac{1}{6}\)
Show your working clearly. (5)
The random variable \(X = 2W - 5\)
Given that \(\mathrm{E}(X) = 2.6\)
| Scheme | Marks | AO |
|---|---|---|
| 2.5 | B1 | 1.1b |
| (1) |
Notes
B1: \(\tfrac{5}{2}\) or 2.5
| Scheme | Marks | AO | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| A complete strategy to find a value for \(a\) and a value for \(b\). | dM1 | 3.1b | ||||||||
| M1 | 2.1 | ||||||||
| \(\tfrac{1}{6}(2a + b) + \tfrac{1}{3}(a + b) = \tfrac{1}{6}\) oe \(4a + 3b = 1\) oe | M1 | 1.1b | ||||||||
| \(3a + 2b = 0.7\) oe | M1 | 1.1b | ||||||||
| \(a = 0.1 \qquad b = 0.2\) | A1 | 1.1b | ||||||||
| (5) |
Notes
dM1: Dependent on 3rd and 4th Method marks being awarded.
For a complete strategy to find a value of \(a\) and a value of \(b\).
Need 2 independent equations in \(a\) and \(b\), one equation must be prob = 1/6 and the other \(3a + 2b = 0.7\) oe and an attempt to solve. For an attempt we require a method to eliminate one variable leading to a value for \(a\) and \(b\), or correct values.
M1: For using the given contextual information to list 3 different combinations for Greg to win.
Implied by \(4a + 3b = 1\) oe
M1: For using \(\mathrm{P}(g) \times \mathrm{P}(n)\) for each combination identified as a win for Greg \(= \tfrac{1}{6}\)
It must be a linear equation in \(a\) and \(b\) with 2/3 terms on the LHS, at least one of which must be correct and equal to 1/6
M1: For use of \(\sum \mathrm{P}(W = w) = 1\)
A1: For both values correct
| Scheme | Marks | AO |
|---|---|---|
| \(2\mathrm{E}(W) - 5 = 2.6\) | M1 | 3.1a |
| \(\mathrm{E}(W) = 3.8\) | ||
| \(8a + b + cb + 1.2\ [= 3.8]\) or \(\text{“}0.3\text{”} + 2 \times \text{“}0.1\text{”} + 4 \times 0.3 + 5 \times \text{“}0.1\text{”} + \text{“}0.2\text{”}c = [\text{“}3.8\text{”}]\) | M1 | 1.1b |
| \(c = \dfrac{2.6 - 8\text{“}a\text{”} - \text{“}b\text{”}}{\text{“}b\text{”}}\) | ||
| \(c = 8\) | A1 | 1.1b |
| \(\mathrm{E}(W^2) = 30a + c^2 b + b + 4.8\) or \(\text{“}0.3\text{”} + 4 \times \text{“}0.1\text{”} + 16 \times 0.3 + 25 \times \text{“}0.1\text{”} + \text{“}8\text{”}^2 \times 0.2\) | M1 | 1.1b |
| \(\mathrm{E}(W^2) = 20.8\) | ||
| \(\mathrm{Var}(W) = \text{“}20.8\text{”} - \text{“}3.8\text{”}^2\) or 6.36 | M1 | 1.1a |
| \(\mathrm{Var}(X) = 25.44\) | A1ft | 1.2 |
| (6) | ||
| (12 marks) |
Notes
M1: For translating the given mathematical context into an expression for \(\mathrm{E}(W)\)
May be implied by a correct equation or \(\mathrm{E}(W) = 3.8\)
M1: For use of \(\sum w\mathrm{P}(W = w)\ [= 3.8]\) If algebraic then at least 2 terms must be correct, if numerical at least 3 terms correct ft their values of \(a\) and \(b\). This must be seen in part (c)
[NB: \(16a + 2b + 2cb + 2.4 - 5 = 2.6\) oe would get M1M1]
A1: cao
M1: For use of \(\sum w^2\mathrm{P}(W = w)\). If algebraic then at least 2 terms must be correct, if numerical at least 3 terms correct ft their values of \(a\) and \(b\).
M1: For use of \(\mathrm{Var}(W) = \mathrm{E}(W^2) - [\mathrm{E}(W)]^2\)
A1ft: \(4 \times \text{“their Var}(W)\text{”}\) ft their \(\mathrm{Var}(W)\) provided \(a\), \(b\) and \(\mathrm{Var}(W)\) are > 0 and \(c \gt 5\)
Alternative for (c). allow a mix of methods
| Scheme | Marks |
|---|---|
| [\(\mathrm{E}(X) =\)] \(-3 \times (a + b) - 1 \times a + 0.9 + 5 \times a + (2c - 5) \times b\) | M1 |
| \(-3 \times (a + b) - 1 \times a + 0.9 + 5 \times a + (2c - 5) \times b = 2.6\) | M1 |
| \(c = 8\) | A1 |
| Values of \(X\) \(-3, -1, 3, 5, 2c - 5\) | M1 |
| \(\mathrm{E}(X^2) = 9 \times (a + b) + 1 \times a + 2.7 + 25 \times a + (2c - 5)^2 \times b\) \(= 32.2\) | M1 |
| \(\mathrm{Var}(X) = \text{“}32.2\text{”} - 2.6^2\) \(= 25.44\) | A1ft |
Notes for alternative
M1: allow with their \(a\), \(b\) and \(X\) values
M1: allow with their \(a\), \(b\) and \(X\) values
A1: cao
M1: at least 3 correct
M1: allow with their \(c\), \(a\), \(b\) and \(X\) values
A1ft: ft their \(\mathrm{E}(X^2)\) provided \(a\) and \(b\) are > 0 and \(c \gt 5\)