S2 June 2011 Q4
4. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm, of a randomly chosen stick has a continuous uniform distribution over the interval [7, 10].
A stick is selected at random from the box.
To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm.
To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm.
| Scheme | Marks |
|---|---|
| \(\dfrac{9.5 - 7}{10 - 7}\) | M1 |
| \(= \dfrac{5}{6}\) awrt 0.833 | A1 |
| (2) |
Notes
M1 for an expression for the probability e.g. \(\displaystyle\int_7^{9.5} \frac{1}{3}\,\mathrm{d}x\)
| Scheme | Marks |
|---|---|
| P(Longest > 9.5) = 1 - P(all < 9.5) \(= 1 - \left(\dfrac{5}{6}\right)^3\) | M1 |
| \(= \dfrac{91}{216}\) or 0.421 | A1 |
| (2) |
Notes
M1 for \(1 - (a)^3\) or \((1 - a)^3 + 3(1 - a)^2 a + 3(1 - a)a^2\)
A1 awrt 0.421
| Scheme | Marks |
|---|---|
| P(a stick < 7.6) \(= \dfrac{0.6}{3} = 0.2\) | B1 |
| Let \(Y\) = number of sticks (out of 6) <7.6 then \(Y \sim \mathrm{B}(6, 0.2)\) \(\mathrm{P}(Y \gt 4) = 1 - \mathrm{P}(Y \leqslant 4)\) | M1 M1 |
| \(= 1 - 0.9984\) \(= 0.0016\) or \(\dfrac{1}{625}\) | A1 |
| (4) | |
| (8 marks) |
Notes
B1 0.2 may be implied by at least one correct probability
1st M1 for writing or using B(6, \(p\)) may be implied by \(np^x(1 - p)^{6-x}\) using their \(p\) and \(n \geqslant 1\)
2nd M1 for writing or using \(1 - \mathrm{P}(Y \leqslant 4)\) or \(np^5(1 - p) + p^6\) (\(n\) is an integer > 1)
A1 cao
NB 0.0016 with no working gets B0M0M0A0