A-LevelMathematicsProbability and statistics9709/53 · Oct/Nov 2022 · Q48 Marks

Three fair 4-sided spinners each have sides labelled 1, 2, 3, 4. The spinners are spun at the same time and the number on the side on which each spinner lands is recorded. The random variable X denotes the highest number recorded. On another occasion, one of the fair 4-sided spinners is spun repeatedly until a 3 is obtained. The random variable Y is the number of spins required to obtain a 3.

(a)Show that P(X = 2) = 7/64.[3]
(b)Complete the probability distribution table for X. X 1 2 3 4 P(X = x) 7/64 19/64[2]
(c)Find P(Y = 6).[1]
(d)Find P(Y > 4).[2]

Mark scheme

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

[B1] (2/4)³ - b seen, 0 < b < 1
[M1] (2/4)³ - c³, 0 < c < 1/2
[A1] Approach identified, WWW.
[B1] P(X=1) or P(X=4) correct. Condone answers not in probability distribution table if clearly identified.
[B1 FT] All 4 probabilities summing to 1.
[B1] Accept 0.059326... to 4 or more SF.
[M1] (3/4)⁴, g=4,5 or p^g where 0 <p<1
[A1] Accept 0.316406...to 4 or more SF.

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