S2 Binomial + Poisson Qs

TheQueen1986

This is Exercise 1G S2 Edexcel. I cannot do d) of these two questions. I can do the rest a) to c). Took a while to type them up, but I think other parts help.

1
a) State the conditions under which the Poisson distribution is a suitable model to use in statistical work.
Flaws in a certain brand of tape occur at random and at an average of 0.75 per 100m. Assuming a Poisson distribution for the average number of flaws in a 400m roll of tape,
b) Find the probability that there will be at least one flaw.
c) Show that the probability that there will be at most 2 flaws is 0.423 (to 3 d.p.)
d) Find the probability that at least 2 rolls will contain fewer than 3 flaws.

3. In Joe's roadside cafe 2/5 of the customers buy a cup of tea.
a) Find the probability that at least 4 of the next 10 customers will buy a cup of tea.
Joe has calculated that on a typical morning customers arrive in the cafe at an average rate of 0.5 per minute.
b) Find the probability that at least 10 customers arrive in the next 15 minutes.
c) Find the probability that exactly 10 customers arrive in the next 20 minutes.
d) Find the probability that in the next 20 minutes exactly 10 customers arrive and at least 4 of them buy a cup of tea.

Reply 1

Jonny W

Find the probability that at least 2 rolls will contain fewer than 3 flaws.

At least two out of how many?

Find the probability that in the next 20 minutes exactly 10 customers arrive and at least 4 of them buy a cup of tea.

The probability of exactly 10 customer arriving is A = e^(-10)(10^10)/10!

Given that exactly 10 customers arrive, the number of cups of tea sold has the Binomial(10, 2/5) distribution. So the probability that at least four cups of tea are sold is

B = 1
- (3/5)^10
- 10*(2/5)(3/5)^9
- (10 choose 2)*(2/5)^2(3/5)^8
- (10 choose 3)*(2/5)^3(3/5)^7

The answer is A * B.