Question

A real estate agent has 1212 properties that she shows. She feels that there is a 50P% chance of selling any one property during a week. The chance of selling any one property is independent of selling another property. Compute the probability of selling no more than 66 properties in one week. Round your answer to four decimal places.

Answer 1

0.6127 = 61.27% probability of selling no more than 6 properties in one week.

Explanation:

For each property, there are only two possible outcomes. Either it is sold, or it is not. The probability of a property being sold is independent of any other property. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

12 properties

This means that
`n = 12`

She feels that there is a 50% chance of selling any one property during a week.

This means that
`p = 0.5`

Compute the probability of selling no more than 6 properties in one week.

This is:

`P(X \leq 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(12,0).(0.5)^(0).(0.5)^(12) = 0.0002`

`P(X = 1) = C_(12,1).(0.5)^(1).(0.5)^(11) = 0.0029`

`P(X = 2) = C_(12,2).(0.5)^(2).(0.5)^(10) = 0.0161`

`P(X = 3) = C_(12,3).(0.5)^(3).(0.5)^(9) = 0.0537`

`P(X = 4) = C_(12,4).(0.5)^(4).(0.5)^(8) = 0.1208`

`P(X = 5) = C_(12,5).(0.5)^(5).(0.5)^(7) = 0.1934`

`P(X = 6) = C_(12,6).(0.5)^(6).(0.5)^(6) = 0.2256`

`P(X \leq 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 0.0002 + 0.0029 + 0.0161 + 0.0537 + 0.1208 + 0.1934 + 0.2256 = 0.6127`

0.6127 = 61.27% probability of selling no more than 6 properties in one week.