Question

Twenty percent of U.S. mortgages are "underwater" (The Boston Globe, March 5, 2009). A mortgage is considered underwater if the value of the home is less than what is owed on the mortgage. Suppose 100 mortgage holders are randomly selected. a. What is the probability that exactly 15 of the mortgages are underwater? (Round your answer to 4 decimal places.)

Answer 1

Answer:

There is a 4.81% probability that exactly 15 of the mortgages are underwater.

Step-by-step explanation:

For each mortgage, there are only two possible outcomes. Either it is "underwater", or it is not. This means that we can solve this problem using concepts of the binomial probability distribution.

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 combinatios of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

In this problem we have that:

Twenty percent of U.S. mortgages are "underwater". This means that $p = 0.20$.

100 mortgage holders are randomly selected. This means that $n = 100$.

a. What is the probability that exactly 15 of the mortgages are underwater?

This is P(X = 15).

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 15) = C_{100,15}.(0.20)^{15}.(0.80)^{85} = 0.0481$

There is a 4.81% probability that exactly 15 of the mortgages are underwater.