June 2018 Paper 3 Q3

EdexcelCurrent spec11 marksBinomial Distribution DRVs Probability

3. In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable \(H\) represents the number of times the dart hits the target in the first 10 throws.

Peta models \(H\) as \(\mathrm{B}(10, 0.1)\)

(a) State two assumptions Peta needs to make to use her model. (2)

(b) Using Peta’s model, find \(\mathrm{P}(H \geqslant 4)\) (1)

For each child the random variable \(F\) represents the number of the throw on which the dart first hits the target.

Using Peta’s assumptions about this experiment,

Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models \(\mathrm{P}(F = n)\) as

\[\mathrm{P}(F = n) = 0.01 + (n - 1) \times \alpha\]

where \(\alpha\) is a constant.

(d) Find the value of \(\alpha\) (4)

(e) Using Thomas’ model, find \(\mathrm{P}(F = 5)\) (1)

(f) Explain how Peta’s and Thomas’ models differ in describing the probability that a dart hits the target in this experiment. (1)

Mark scheme (a)

Scheme	Marks	AO
The probability of a dart hitting the target is constant (from child to child and for each throw by each child) (o.e.)	B1	1.2
The throws of each of the darts are independent (o.e.)	B1	1.2
	(2)

Notes

1^st B1 for stating that the probability (or possibility or chance) is constant (or fixed or same)

2^nd B1 for stating that throws are independent [“trials” are independent is B0]

Scheme	Marks	AO
\([\mathrm{P}(H \geqslant 4) = 1 - \mathrm{P}(H \leqslant 3) = 1 - 0.9872 = 0.012795\ldots =]\) awrt 0.0128	B1	1.1b
	(1)

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{P}(F = 5) = 0.9^4 \times 0.1,\ = 0.06561\) \(=\) awrt 0.0656	M1, A1	3.4 1.1b
	(2)

Notes

M1 for a probability expression of the form \((1 - p)^4 \times p\) where \(0 \lt p \lt 1\)

A1 for awrt 0.0656

SC Allow M1A0 for answer only of 0.066

Mark scheme (d)

Scheme

Marks

\(n\)	\(1\)	\(2\)	\(\ldots\)	\(10\)
\(\mathrm{P}(F = n)\)	\(0.01\)	\(0.01 + \alpha\)	\(\ldots\)	\(0.01 + 9\alpha\)

3.1b

Sum of probs \(= 1 \quad \Rightarrow \dfrac{10}{2}[2 \times 0.01 + 9\alpha] = 1\)

M1A1

3.1a
1.1b

[i.e. \(5(0.02 + 9\alpha) = 1\) or \(0.1 + 45\alpha = 1\)] so \(\alpha = \mathbf{\underline{0.02}}\)

1.1b

(4)

Notes

1^st M1 for setting up the distribution of \(F\) with at least 3 correct values of \(n\) and \(\mathrm{P}(F = n)\) in terms of \(\alpha\). (Can be implied by 2^nd M1 or 1^st A1)

2^nd M1 for use of sum of probs = 1 and clear summation or use of arithmetic series formula (allow 1 error or missing term). (Can be implied by 1^st A1)

1^st A1 for a correct equation for \(\alpha\)

2^nd A1 for \(\alpha = 0.02\) (must be exact and come from correct working)

Mark scheme (e)

Scheme	Marks	AO
\(\mathrm{P}(F = 5 \mid \text{Thomas’ model}) = \mathbf{\underline{0.09}}\)	B1ft	3.4
	(1)

Notes

B1ft for value resulting from \(0.01 + 4 \times \text{``}\text{their } \alpha\text{''}\) (provided \(\alpha\) and the answer are probs)

Beware If their answer is the same as their (c) (or a rounded version of their (c)) score B0

Mark scheme (f)

Scheme	Marks	AO
Peta’s model assumes the probability of hitting target is constant (o.e.) and Thomas’ model assumes this probability increases with each attempt (o.e.)	B1	3.5a
	(1)
	(11 marks)

Notes

B1 for a suitable comment about the probability of hitting the target

ALT Allow idea that Peta’s model suggests the dart may never hit the target but Thomas’ says that it will hit at least once (in the first 10 throws).