S2 June 2009 Q1

EdexcelOld spec5 marksPoisson Distribution

1. A bag contains a large number of counters of which 15% are coloured red. A random sample of 30 counters is selected and the number of red counters is recorded.

(a) Find the probability of no more than 6 red counters in this sample. (2)

A second random sample of 30 counters is selected and the number of red counters is recorded.

(b) Using a Poisson approximation, estimate the probability that the total number of red counters in the combined sample of size 60 is less than 13. (3)

Mark scheme (a)

Scheme	Marks
\(\left[X \sim \mathrm{B}(30, 0.15)\right]\)
\(\mathrm{P}(X \leqslant 6),\ = 0.8474\) awrt 0.847	M1, A1
	(2)

Notes

M1 for a correct probability statement \(\mathrm{P}(X \leqslant 6)\) or \(\mathrm{P}(X \lt 7)\) or \(\mathrm{P}(X = 0) + \mathrm{P}(X = 1) + \mathrm{P}(X = 2) + \mathrm{P}(X = 3) + \mathrm{P}(X = 4) + \mathrm{P}(X = 5) + \mathrm{P}(X = 6)\). (may be implied by long calculation)
Correct answer gets M1 A1. allow 84.74%

(corrected from the printed mark scheme: the printed sum leaves out \(\mathrm{P}(X = 3)\))

Mark scheme (b)

Scheme	Marks
\(Y \sim \mathrm{B}(60, 0.15) \quad \approx \quad \mathrm{Po}(9)\) for using Po(9)	B1
\(\mathrm{P}(Y \leqslant 12),\ = 0.8758\) awrt 0.876	M1, A1
[ N.B. normal approximation gives 0.897, exact binomial gives 0.894]
	(3)
	(5 marks)

Notes

B1 may be implied by using Po(9). Common incorrect answer which implies this is 0.9261

M1 for a correct probability statement \(\mathrm{P}(X \leqslant 12)\) or \(\mathrm{P}(X \lt 13)\) or \(\mathrm{P}(X = 0) + \mathrm{P}(X = 1) + \ldots + \mathrm{P}(X = 12)\) (may be implied by long calculation) and attempt to evaluate this probability using their Poisson distribution.

Condone \(\mathrm{P}(X \leqslant 13) = 0.8758\) for B1 M1 A1

Correct answer gets B1 M1 A1

Use of normal or exact binomial get B0 M0 A0