June 2024 Paper 2 Q8

OCR ACurrent spec6 marksBinomial Distribution

8 Sweets from a certain manufacturer are sold in packets. Thirty per cent of the sweets are orange, and these are randomly distributed amongst the packets. Each packet contains 15 sweets.

The number of orange sweets in a randomly chosen packet is denoted by \(X\).

(a) Find the following probabilities.

(i) \(\mathrm{P}(X = 4)\) [1]

(ii) \(\mathrm{P}(X \geqslant 4)\) [2]

(b)

(i) Write down an expression for \(\mathrm{P}(X = r)\). [1]

(ii) Explain the connection between the expression in part (b)(i) and the binomial expansion of \((0.7 + 0.3)^n\), for a specific value of \(n\) which should be stated. [2]

Mark scheme (a)

Scheme	Marks	AO
(i) 0.219 (3 sf)	B1	1.1
	[1]
(ii) \(1 - \mathrm{P}(X \leqslant 3)\) or \(1 - 0.297\)	M1	3.4
= 0.703 (3 sf)	A1	1.1
	[2]

Notes

(a)(i) B1: awrt 0.219 (Condone 21.9%)

(a)(ii)

M1: Allow M1 for \(1 - \mathrm{P}(X \leqslant 4)\) or \(1 - 0.515\) or 0.484
May be implied by correct answer.

A1: awrt 0.703 (Condone 70.3%)

Mark scheme (b)

Scheme	Marks	AO
(i) \(\dbinom{15}{r}0.7^{15-r}0.3^r\) oe	B1	1.2
	[1]
(ii) \(n = 15\)	B1	3.3
\(\mathrm{P}(X = r)\) is the \((r \pm 1)\)th term of the expansion	B1	2.4
	[2]

Notes

(b)(i) B1: Allow \((1 - 0.3)\) for 0.7
Accept \({}^{15}\mathrm{C}_r\) etc. but must be in terms of \(r\)

(b)(ii)

B1: Must be stated (not implied)

B1: Allow \(\mathrm{P}(X = r)\) is a term of the expansion
Condone “appears in the expansion” but not “it is the expansion” and not “the coefficient of a term”
Condone \(\mathrm{P}(X = r)\) is the \(r\)th term of the expansion