A-LevelMathematicsProbability and statistics9709/51 · May/June 2022 · Q410 Marks

Jacob has four coins. One of the coins is biased such that when it is thrown the probability of obtaining a head is 7/10. The other three coins are fair. Jacob throws all four coins once. The number of heads that he obtains is denoted by the random variable X. The probability distribution table for X is as follows. X | 0 | 1 | 2 | 3 | 4 P(X = x) | 3/80 | a | b | c | 7/80

(a)Show that a = 1/5 and find the values of b and c.[4]
(b)Find E(X).[1]
(c)Jacob throws all four coins together 10 times. Find the probability that he obtains exactly one head on fewer than 3 occasions.[3]
(d)Find the probability that Jacob obtains exactly one head for the first time on the 7th or 8th time that he throws the 4 coins.[2]

Mark scheme

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

[B1] Clear statement of unevaluated correct calculation = 1/5 AG
[M1] Clear statement of unevaluated calculation for either b or c
[A1] For either b or c correct
[B1 FT] their b + their c = 27/40
[B1 FT] Correct or accept unsimplified calculation using their values for b and c seen (sum of probabilities = 1)
[M1] One term 10Cx p^x (1-p)^(10-x), for 0<x<10, 0<p<1
[A1] Correct expression, accept unsimplified leading to final answer
[B1] 0.677 < p <= 0.678
[M1] p^l x (1-p) + p^m x (1-p), l = 6, 7 m = l+1, 0<p<1
[A1] 0.09437 < p < 0.0944

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