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Learn/ A Level Mathematics/ Past Papers/ 2021/ May/June 2021 /Paper 6 (Variant 1)

Probability & Statistics 2 — Variant 1

A Level Mathematics · 9709/61 · May/June 2021

Question Paper

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Questions

7 of 8 questions shown here, checked against the paper. The others are in the PDF above — they contain diagrams or maths we could not yet verify, and we would rather show you the original than a transcription we are not sure of.

1 [4]
Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.13.1 at factory AA and 1.71.7 at factory BB. Find the probability that the total number of accidents in the two factories during a 2-month period is more than 33.
2
The time, in minutes, taken by students to complete a test has the distribution N(125,36)\mathrm{N}(125, 36).
2(a) [3]
Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123123 minutes.
2(b) [1]
Explain whether it was necessary to use the Central Limit theorem in the solution to part (a).
3 [2]
The graph of the probability density function of a random variable XX is symmetrical about the line x=4x = 4. Given that P(X<5)=2027\mathrm{P}(X < 5) = \frac{20}{27}, find P(3<X<5)\mathrm{P}(3 < X < 5).
4 [6]
100 randomly chosen adults each throw a ball once. The length, ll metres, of each throw is recorded. The results are summarised below. n=100n = 100 Σl=3820\Sigma l = 3820 Σl2=182200\Sigma l^{2} = 182 200 Calculate a 94% confidence interval for the population mean length of throws by adults.
5
On average, 1 in 75 000 adults has a certain genetic disorder.
5(a) [3]
Use a suitable approximating distribution to find the probability that, in a random sample of 10 000 people, at least 11 has the genetic disorder.
5(b) [4]
In a random sample of nn people, where nn is large, the probability that no-one has the genetic disorder is more than 0.90.9. Find the largest possible value of nn.
6 [6]
The probability density function, ff, of a random variable XX is given by f(x)={k(6x−x2)0⩽x⩽6,0otherwise,f(x) = \begin{cases} k(6x - x^{2}) & 0 \leqslant x \leqslant 6, \\ 0 & \text{otherwise,} \end{cases} where kk is a constant. State the value of E(X)\mathrm{E}(X) and show that Var(X)=95\mathrm{Var}(X) = \frac{9}{5}.
7
The masses, in kilograms, of large and small sacks of flour have the distributions N(55,32)\mathrm{N}(55, 3^{2}) and N(27,2.52)\mathrm{N}(27, 2.5^{2}) respectively.
7(a) [5]
Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340340 kg. Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour.
7(b) [5]
Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour.

Published by Cambridge Assessment International Education. Reproduced for study use.