A-LevelMathematicsProbability and statistics9709/62 · Feb/Mar 2022 · Q611 Marks

In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable X with probability density function f(x) = { -k(x - 1)(x - 3) 1 ≤ x ≤ 3, {; lo 0 otherwise, where k is a constant.

(a)Without calculation, explain why E(X) = 2.[1]
(b)Show that k = 3/4.[3]
(c)Find Var(X).[3]
(d)One turn consists of rolling the ball 3 times and noting the largest value of X obtained. If this largest value is greater than 2.5, the player scores a point. Find the probability that on a particular turn the player scores a point.[4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2022). The mark or mark type is shown in brackets.

[B1] Quadratic curve, hence symmetrical OE. Allow sketch and 'symmetrical' or just 'curve symmetrical'
[M1] Attempt to integrate f(x) and ‘= 1'. Ignore limits at this stage
[A1] -k[x³/3 - 2x² + 3x]³₁ Fully correct expression (correct integration and limits)
[A1] -k[ (27/3 - 18 + 9) - (1/3 - 2 + 3) ] = 1 or k x 4/3 = 1 AG, OE. Correctly substitute limits and ‘= 1' and correctly obtain result with no errors seen. k = 3/4
[M1] Attempt to integrate x² f(x) from 1 to 3
[A1] (3/4) * [x⁵/5 - x⁴ + x³]³₁ Correct integration and limits
[A1] = 0.2
[M1] OE. Attempt to integrate f(x), from 2.5 to 3 (or 1 to 2.5)
[A1] = - (3/4) * [x³/3 - 2x² + 3x]³₂.₅ or 5/32 or 0.15625
[M1] OE. FT their 5/32 1 - (1 - 5/32)³
[A1] = 0.399 (3 sf)

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