S2 June 2012 Q8

EdexcelOld spec8 marksBinomial Distribution

8. In a large restaurant an average of 3 out of every 5 customers ask for water with their meal.

A random sample of 10 customers is selected.

(a) Find the probability that

(i) exactly 6 ask for water with their meal,

(ii) less than 9 ask for water with their meal.

(5)

A second random sample of 50 customers is selected.

(b) Find the smallest value of \(n\) such that \[\mathrm{P}(X \lt n) \geqslant 0.9\] where the random variable \(X\) represents the number of these customers who ask for water. (3)

Mark scheme (a)

Scheme	Marks
Let \(X\) be the random variable the number of customers asking for water.
(i) \(X \sim \mathrm{B}(10, 0.6)\) or \(Y \sim \mathrm{B}(10, 0.4)\)	B1
\(\mathrm{P}(X = 6) = (0.6)^6(0.4)^4\dfrac{10!}{6!4!}\) or \(\mathrm{P}(Y = 4) = (0.4)^4(0.6)^6\dfrac{10!}{6!4!}\)	M1
\(= 0.2508\ldots\) \(= 0.2508\) awrt 0.251	A1
(ii) \(X \sim \mathrm{B}(10, 0.6)\): \(\mathrm{P}(X \lt 9) = 1 - (\mathrm{P}(X = 10) + \mathrm{P}(X = 9))\) \(= 1 - (0.6)^{10} - (0.6)^9(0.4)^1\dfrac{10!}{9!1!}\) or \(Y \sim \mathrm{B}(10, 0.4)\): \(\mathrm{P}(X \lt 9) = 1 - \mathrm{P}(Y \leqslant 1)\) \(= 1 - 0.0464\)	M1
\(= 0.9536\ldots\) \(= 0.9536\ldots\) awrt 0.954	A1
	(5)

Notes

B1 writing or using B(10,0.6) / B(10,0.4) in either part(i) or (ii)

(i) M1 \((0.6)^6(1 - 0.6)^4\dfrac{10!}{6!4!}\) Allow \({}^{10}\mathrm{C}_6\) oe or writing or using \(\mathrm{P}(X \leqslant 6) - \mathrm{P}(X \leqslant 5)\) if using B(10,0.6) or \(\mathrm{P}(X \leqslant 4) - \mathrm{P}(X \leqslant 3)\) if using B(10,0.4)

NB use of Poisson will gain M0A0

(ii) M1 writing or using \(1 - (\mathrm{P}(X = 10) + \mathrm{P}(X = 9))\) if using B(10,0.6) or \(1 - \mathrm{P}(Y \leqslant 1)\) if using B(10,0.4)

NB use of Poisson will gain M0A0

Mark scheme (b)

Scheme	Marks
\(X \sim \mathrm{B}(50, 0.6)\) \(Y \sim \mathrm{B}(50, 0.4)\)	M1
\(\mathrm{P}(X \lt n) \geqslant 0.9\) \(\mathrm{P}(Y \gt 50 - n) \geqslant 0.9\) or \(\mathrm{P}(X \lt 34) = 0.8439\) awrt 0.844 \(\mathrm{P}(Y \leqslant 50 - n) \leqslant 0.1\) \(\mathrm{P}(X \lt 35) = 0.9045\) awrt 0.904/0.905	M1
\(50 - n \leqslant 15\) \(n \geqslant 35\) \(n = 35\)	A1
	(3)
	(8 marks)

Notes

1st M1 for writing or using either B(50,0.6) or B(50,0.4)

2nd M1 \(\mathrm{P}(Y \gt 50 - n) \geqslant 0.9\) or \(\mathrm{P}(Y \leqslant 50 - n) \leqslant 0.1\) or \(\mathrm{P}(X \lt 34) =\) awrt 0.844 or \(\mathrm{P}(X \lt 35) =\) awrt 0.904/0.905 or \(50 - n = 15\) or \(50 - n = 16\) or \(50 - n \leqslant 15\) or \(50 - n \leqslant 16\) – allow different letters

A1 cao 35. Do not accept \(n \geqslant 35\) for final A1.

SC use of normal.

M1 M0 A0 for use of N(30,12) leading to an answer of 35