Higher June 2022 Paper 1 Q15
15 Abraham is going to play a computer game.
Abraham can win the game, draw the game or lose the game.
For any game that Abraham plays
the probability that he wins the game is 0.3
the probability that he draws the game is 0.5
the probability that he loses the game is 0.2
When Abraham wins a game, he scores +10 points.
When Abraham draws a game, he scores 0 points.
When Abraham loses a game, he scores −5 points.
Abraham plays 3 games and the points he scores in each of the 3 games are added together to get his total score.
Work out the probability that when he has played 3 games his total score is 0 points.
(4)
| Scheme | Marks |
|---|---|
| \(0.5^3\) or \(\dfrac{1}{8}\) or 0.125 oe | M1 |
| \(0.3 \times 0.2^2\) or \(\dfrac{3}{250}\) or 0.012 oe | M1 |
\(0.5^3 + 3 \times 0.3 \times 0.2^2\) or \(\text{``}{\dfrac{1}{8}}\text{''} + \text{``}{\dfrac{9}{250}}\text{''}\) or “0.125” + 3 × “0.012” oe | M1 |
| 0.161 | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for finding \(DDD\)
M1: for finding \(WLL\) in any order
M1: for a complete method
A1: oe
| Scheme | Marks |
|---|---|
| \(0.3^3\) or 0.027 or \(0.2^3\) or 0.008 oe | M1 |
| \(0.3^2 \times 0.5\) or 0.045 or \(0.3^2 \times 0.2\) or 0.018 or \(0.5^2 \times 0.3\) or 0.075 or \(0.5^2 \times 0.2\) or 0.05 or \(0.2^2 \times 0.5\) or 0.02 or \(0.3 \times 0.5 \times 0.2\) or 0.03 or \(0.3^2 \times 0.7\) or 0.063 or \(0.5^2 \times 0.5\) or 0.125 or \(0.2^2 \times 0.5\) or 0.02 or \(0.3 \times 0.5 \times 0.2\) or 0.03 | M1 |
| \(1 - (3 \times 0.3^2 \times 0.5 + 3 \times 0.3^2 \times 0.2 + 3 \times 0.5^2 \times 0.3 + 3 \times 0.5^2 \times 0.2 + 3 \times 0.2^2 \times 0.5 + 6 \times 0.3 \times 0.5 \times 0.2)\) or \(1 - (3 \times 0.3^2 \times 0.7 + 3 \times 0.5^2 \times 0.5 + 3 \times 0.2^2 \times 0.5 + 6 \times 0.3 \times 0.5 \times 0.2)\) | M1 |
| 0.161 | A1 |
Notes
M1: for finding \(WWW\) or \(LLL\)
M1: for finding \(WWD\) or \(WWL\) or \(WDD\) or \(DDL\) or \(DLL\) or \(WDL\) in any order
or
for finding \(WWW\)′ or \(DDD\)′ or \(DLL\) or \(WDL\) in any order
M1: for a complete method
A1: oe