S2 June 2010 Q2
2. Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2
Find the probability that, in 9 games, Bhim loses
Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05
Bhim and Joe agree to play a further 60 games.
| Scheme | Marks |
|---|---|
| Let \(X\) be the random variable the number of games Bhim loses. \(X \sim \mathrm{B}(9, 0.2)\) | B1 |
| \(\mathrm{P}(X \leqslant 3) - \mathrm{P}(X \leqslant 2) = 0.9144 - 0.7382\) or \((0.2)^3(0.8)^6\dfrac{9!}{3!6!}\) | M1 |
| \(= 0.1762\) \(= 0.1762\) awrt 0.176 | A1 |
| (3) |
Notes
B1 – writing or use of B(9, 0.2)
M1 for writing/ using \(\mathrm{P}(X \leqslant 3) - \mathrm{P}(X \leqslant 2)\) or \((p)^3(1 - p)^6\dfrac{9!}{3!6!}\)
A1 awrt 0.176
Special case :Use of Po(1.8) in (a) and (b)
(a) can get B1 M1 A0 – B1 if written B(9, 0.2), M1 for \(\dfrac{\mathrm{e}^{-1.8}1.8^3}{3!}\) or awrt to 0.161
If B(9, 0.2) is not seen then the only mark available for using Poisson is M1.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \leqslant 4) = 0.9804\) awrt 0.98 | M1A1 |
| (2) |
Notes
M1 for writing or using \(\mathrm{P}(X \leqslant 4)\)
A1 awrt 0.98
Special case :Use of Po(1.8) in (a) and (b)
(b) can get M1 A0 - M1 for writing or using \(\mathrm{P}(X \leqslant 4)\) or may be implied by awrt 0.964
| Scheme | Marks |
|---|---|
| Mean = 3 variance = 2.85, \(\dfrac{57}{20}\) | B1 B1 |
| (2) |
Notes
B1 3
B1 2.85, or exact equivalent
| Scheme | Marks |
|---|---|
| Po(3) poisson | M1 |
| \(\mathrm{P}(X \gt 4) = 1 - \mathrm{P}(X \leqslant 4)\) | M1 |
| \(= 1 - 0.8153\) \(= 0.1847\) | A1 |
| (3) | |
| (10 marks) |
Notes
M1 for using Poisson
M1 for writing or using \(1 - \mathrm{P}(X \leqslant 4)\) NB \(\mathrm{P}(X \leqslant 4)\) is 0.7254 Po(3.5) and 0.8912 Po(2.5)
A1 awrt 0.185
Use of Normal in (d)
Can get M0 M1 A0.- for M1 they must write \(1 - \mathrm{P}(X \leqslant 4)\) or get awrt 0.187