S2 June 2010 Q3
3. A rectangle has a perimeter of 20 cm. The length, \(X\) cm, of one side of this rectangle is uniformly distributed between 1 cm and 7 cm.
Find the probability that the length of the longer side of the rectangle is more than 6 cm long. (5)
| Scheme | Marks |
|---|---|
| Method 1 \(\mathrm{P}(X \gt 6) = \dfrac{1}{6}\) | B1 M1 |
| \(\mathrm{P}(X \lt 4) = \dfrac{1}{2}\) | A1 |
| \(\text{total} = \dfrac{1}{6} + \dfrac{1}{2} = \dfrac{2}{3}\) | M1dep B A1 |
| (5) | |
| (5 marks) |
Notes
Method 2
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(4 \lt X \lt 6) = \dfrac{1}{3}\) | B1 M1 A1 |
| \(1 - \dfrac{1}{3} = \dfrac{2}{3}\) | M1dep B A1 |
Method 3
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(X \gt 6) = \dfrac{1}{6}\) | B1 M1 |
| \(Y \sim \mathrm{U}[3,9]\) \(\mathrm{P}(Y \gt 6) = \dfrac{1}{2}\) | A1 |
| \(\text{total} = \dfrac{1}{6} + \dfrac{1}{2} = \dfrac{2}{3}\) | M1dep B A1 |
Methods 1 and 2
B1 for 6 and 4 (allow if seen on a diagram on \(x\)-axis)
M1 for \(\mathrm{P}(X \gt 6)\) or \(\mathrm{P}(6 \lt X \lt 7)\); or \(\mathrm{P}(X \lt 4)\) or \(\mathrm{P}(1 \lt X \lt 4)\) ; or \(\mathrm{P}(4 \lt X \lt 6)\) Allow \(\leqslant\) and \(\geqslant\) signs
A1 \(\dfrac{1}{6}\); or \(\dfrac{1}{2}\); \(\dfrac{1}{3}\) must match the probability statement
M1 for adding their “\(\mathrm{P}(X \gt 6)\)” and their “\(\mathrm{P}(X \lt 4)\)” or 1 - their “\(\mathrm{P}(4 \lt X \lt 6)\)” dep on getting first B mark
A1 cao \(\dfrac{2}{3}\)
Method 3 \(Y \sim \mathrm{U}[3, 9]\)
B1 for 6 with U[1,7]and 6 with U[3,9]
M1 for \(\mathrm{P}(X \gt 6)\) or \(\mathrm{P}(6 \lt X \lt 7)\) or \(\mathrm{P}(6 \lt Y \lt 9)\)
A1 \(\dfrac{1}{6}\); or \(\dfrac{1}{2}\); must match the probability statement
M1 for adding their “\(\mathrm{P}(X \gt 6)\)” and their “\(\mathrm{P}(Y \gt 6)\)” dep on getting first B mark
A1 cao \(\dfrac{2}{3}\)