A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2017 · Q47 Marks

A fair tetrahedral die has faces numbered 1, 2, 3, 4. A coin is biased so that the probability of showing a head when thrown is 1/3. The die is thrown once and the number n that it lands on is noted. The biased coin is then thrown n times. So, for example, if the die lands on 3, the coin is thrown 3 times.

(i)Find the probability that the die lands on 4 and the number of times the coin shows heads is 2.[3]
(ii)Find the probability that the die lands on 3 and the number of times the coin shows heads is 3.[1]
(iii)Find the probability that the number the die lands on is the same as the number of times the coin shows heads.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

M1 Multiplying their 2H expression by ¼ [P(4)]
M1 Remaining factor is (1/3)²(2/3)² [or 4/81] multiplied by integer value k≥ 1 OE
A1 = 2/27 (0.0741)
B1 P(3,3H) = 1/4 x (1/3)³ = 1/108 (0.00926)
M1 Correct expression for 1 of P(1, 1H), P(2, 2H), P(4, 4H) Unsimplified (or better)
M1 Summing their values for 3 or 4 appropriate outcomes for the 'game' with no additional outcomes.
A1 Prob = 10/81 (0.123)

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