Question

Suppose my grocery store of choice has 20 shopping carts in total and 10 of them have wobbly wheels (they don’t really push properly). I’ve gone to the store with 3 of my friends to stock up for a big party and we decide to split up (each of us taking a cart, so 4 carts in total). Let X be the number of carts with wobbly wheels (out of the 4 me and my friends have taken).What is the distribution of X?Give the name of the distribution and the appropriate parameters.What is the mean and variance of this distribution? Give your answer as a number,but include the formulas (or logic) used.

What is the probability exactly 2 of us have carts with wobbly wheels?

What is the probability that at least 1 of us has a cart with wobbly wheels?

Answer 1

Here X denotes the number of carts with wobbly wheels out of the 4 taken. The distribution of X is binomial distribution as it follows the following requirements for a binomial distribution.

1) There must be n independent and identical trials.

2) There is a non-zero constant probability of success.

3) There must be two outcomes for the variable.

Here the possible two outcomes of X are cart with wobbly wheels (assumed as success), and cart without wobbly wheels (assumed as failure). The selection of the carts are independent to each other and random. The probablity of getting cart with wobbly wheels is 10 / 20 = 0.5. So X has a binomial distribution with parameters n = 4 and           p = 0.5. The probability distribution for X is given as

for x = 0, 1, 2, 3 and 4.

The mean of the distribution is given as np = 4(0.5) = 2

The variance of the distribution is given as np(1-p) = 4(0.5)(0.5) = 1

The probability that exactly 2 of us have carts with wobbly wheels is given as .

The probability exactly 2 get the wobbly wheels is 0.375.

The probability that at least 1 of 4 has a cart with wobbly wheels is given as

The probability that at least 1 of 4 has a cart with wobbly wheels is 0.9375.