S2 June 2013 Q7

EdexcelOld spec10 marksBinomial Distribution

7. A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1

(a) Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day. (1)

(b) On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines. (3)

(c) Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95 (3)

A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05

The call centre telephones 100 people every hour.

(d) Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour. (3)

Scheme	Marks
Distribution \(X \sim \mathrm{B}(n, 0.1)\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(Y \sim \mathrm{B}(10, 0.1)\)	B1
\(\mathrm{P}(Y \geqslant 4) = 1 - \mathrm{P}(Y \leqslant 3)\)	M1
\(= 1 - 0.9872\)
\(= 0.0128\)	A1
	(3)

Notes

B1 writing or using B(10,0.1)

M1 writing or using \(1 - \mathrm{P}(Y \leqslant 3)\)

A1 awrt 0.0128

Mark scheme (c)

Scheme	Marks
\(0.9^n \lt 0.05\) or \(1 - (0.9)^n \gt 0.95\)	M1
\(n \gt 28.4\)	A1
\(n = 29\)	A1
	(3)

Notes

Alternative

Scheme	Marks
B(28,0.1): P(0) = 0.0523	M1
B(29,0.1): P(0) = 0.0471	A1
\(n = 29\)	A1cao
	(3)

M1 \((0.9)^n \lt 0.05\),oe, or \((0.9)^n = 0.05\),oe, or \((0.9)^n \gt 0.05\), oe, or seeing 0.0523 or seeing 0.0471

1st A1 [P(0)] = 0.0471 or getting awrt 28.4 May be implied by correct answer.

2nd A1 cao \(n = 29\) should not come from incorrect working.

NB An answer of 29 on its own with no working gains M1A1A1

Mark scheme (d)

Scheme	Marks
\(C \sim \mathrm{Po}(5)\)	B1
\(\mathrm{P}(C \gt 10) = 1 - \mathrm{P}(C \leqslant 10)\)	M1
\(= 1 - 0.9863\)
\(= 0.0137\)	A1
	(3)
	(10 marks)

Notes

B1 writing or using Po(5)

M1 writing or using \(1 - \mathrm{P}(C \leqslant 10)\)

A1 awrt 0.0137