A-LevelMathematicsProbability and statistics9709/61 · May/June 2020 · Q510 Marks

Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per match. In their away matches they score goals at a constant average rate of 0.8 goals per match. You may assume that goals are scored at random times and independently of one another.

(a)(i)Find the probability that the team scores a total of 4 goals in their two matches.[2]
(a)(ii)Find the probability that the team scores a total of 4 goals, with more goals scored in the home match than in the away match.[3]
(b)Use a suitable approximating distribution to find the probability that the team scores fewer than 25 goals in 10 randomly chosen weeks.[4]
(c)Justify the use of the approximating distribution used in part (b).[1]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[M1] e⁻²⋅⁹ × 2.9⁴ / 4!
[A1] 0.162 (3 sf)
[B1] e⁻²⋅¹ × 2.1⁴ / 4! × e⁻⁰⋅⁸ + e⁻²⋅¹ × 2.1³ / 3! × e⁻⁰⋅⁸ × 0.8 + e⁻²⋅¹ × 2.1² / 2! × e⁻⁰⋅⁸ × 0.8² / 2!
[M1] for P(4, 0) + P(3, 1) + P(2, 2) where home goals > away goals
[A1] 0.113 (3 sf)
[M1] N(29, 29)
[M1] (24.5-29) / √29 (=-0.83563)
[M1] 1 – Ф("0.836")
[A1] 0.202 (3sf)
[B1] 29 is large or 29 > 15

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.