Question

The probability that a person living in a certain city owns a bicycle is estimated to be 0.3.

(a) If 10 persons are randomly selected, find the probability that there are exactly three persons own a bicycle.
(b) Find the probability that the tenth person randomly interviewed in this city is the
(i) first one, (ii)fifth one
to own a bicycle.
Question 2
(6 marks)
In a large factory, the number of defective items produced each day could be modelled by a Poisson distribution with mean λ. The number of defective items produced daily over 100 working days is tabulated as follows to estimate the mean λ.
5 or above
Observed 20 37 22 12 9 0 frequency (days)
(a) Find the mean of observed data.
(b) Using the answer in (a), find the probability that
(i) no defective item is produced in a day.
(ii) more than one defective item is produced in a day.
(iii) in two days out of five consecutive days, exactly 3 defective items are produced each day.
Number of defective items produced
0
1
2
3
4

Answer 1

a) The probability that exactly three persons own a bicycle out of 10 randomly selected persons is approximately 0.2668.

bi) The probability that the tenth person interviewed is the first one to own a bicycle is approximately 0.0024.

bii) The probability that the tenth person interviewed is the fifth one to own a bicycle is approximately 0.0006.

How the probabillities are determined:

a) Let the number of persons who own a bicycle out of 10 randomly selected persons = X

The probability that a person living in the city owns a bicycle = 0.3

Using a binomial random variable with the following parameters:

n = 10

p = 0.3.

The probability that exactly three persons own a bicycle is given by the probability mass function of X evaluated at k = 3:

P(X = 3) = (10 choose 3) * (0.3)^3 * (0.7)^7 ≈ 0.2668

Thus, the probability that exactly three persons own a bicycle out of 10 randomly selected persons is approximately 0.2668.

b) Let the number of persons who own a bicycle out of the first 10 persons interviewed = Y

The probability that a person living in the city owns a bicycle = 0.3

Using a binomial random variable with parameters:

n = 10

p = 0.3

The probability that the tenth person interviewed is the first one to own a bicycle is given by the probability that the first nine persons do not own a bicycle and the tenth person does own a bicycle:

P(Y = 1) = (0.7)^9 * (0.3) ≈ 0.0024

Thus, the probability that the tenth person interviewed is the first one to own a bicycle is approximately 0.0024.

ii) Similarly, the probability that the tenth person interviewed is the fifth one to own a bicycle is given by the probability that the first four persons do not own a bicycle, the next four persons do not own a bicycle, and the tenth person does own a bicycle:

P(Y = 5) = (0.7)^8 * (0.3)^2 * (0.7)^1 ≈ 0.0006

Therefore, the probability that the tenth person interviewed is the fifth one to own a bicycle is approximately 0.0006.