Practise calculating Poisson probabilities, rescaling the mean across time or space and choosing Poisson or normal approximations with conditions and continuity corrections.
The number, X, of books received at a charity shop has a constant mean of 5.1 per day.
State, in context, one condition for X to be modelled by a Poisson distribution.
Assume now that X can be modelled by a Poisson distribution.
Books received independently or singly or randomly.
B1
OE. Must be in context.
If more than one condition given, ignore extras.
Find the probability that exactly 10 books are received in a 3-day period.
e^-15.3 x 15.3^1010!
M1
Allow incorrect .
=0.0439(3 sf)
A1
SC No working shown but correct answer seen scores
B1.
Use a suitable approximating distribution to find the probability that more than 180 books are received in a 30-day period.
The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean 2.5 per day.
N(153, 153)
B1
Seen or implied.
180.5-153153 [=2.223]
M1
For standardising with their values (can be implied).
Allow with wrong or missing continuity correction.
1- ('2.223')
M1
For correct probability area consistent with their values.
=0.0131(3 sf)
A1
The number of absences per week by workers at a factory has the distribution Po(2.1).
Find the standard deviation of the number of absences per week.
2.1 or 1.45(3 sf)
B1
Find the probability that the number of absences in a 2-week period is at least 2 .
λ=4.2
B1.
1−e−4.2(1+4.2)
M1 1−P(X⩽1); allow one end error.
A1 0.922 (3 s.f.).
Find the probability that the number of absences in a 3-week period is more than 4 and less than 8 .
Following a change in working conditions, the management wished to test whether the mean number of absences has decreased. They found that, in a randomly chosen 3-week period, there were exactly 2 absences.
λ=6.3e−6.3(5!6.35+6!6.36+7!6.37)
M1 P(X=5,6,7); allow one end error.
A1 0.455 (3 s.f.).