In finance, one example of a derivative is a financial asset whose value is determined (derived) from a bundle of various assets, such as mortgages. Suppose a randomly selected mortgage in a certain bundle has a probability of 0.08 of default. (a) What is the probability that a randomly selected mortgage will not default? (b) What is the probability that nine randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others? Note: A derivative might be an investment that only pays when all nine mortgages do not default. (c) What is the probability that the derivative from part (b) becomes worthless? That is, at least one of the mortgages defaults

a) 92% probability that a randomly selected mortgage will not default

b) 47.22% probability that nine randomly selected mortgages will not default

c) 52.78% probability that the derivative from part (b) becomes worthless

Step-by-step explanation:

Binomial probability distribution

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

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

And p is the probability of X happening

Suppose a randomly selected mortgage in a certain bundle has a probability of 0.08 of default.

This means that

(a) What is the probability that a randomly selected mortgage will not default?

Either it defaults, or it does not default. The sum of the probabilities of these outcomes is 1. So

0.08 + p = 1

p = 0.92

92% probability that a randomly selected mortgage will not default

(b) What is the probability that nine randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others?

This is P(X = 0) when n = 9. So

47.22% probability that nine randomly selected mortgages will not default.

(c) What is the probability that the derivative from part (b) becomes worthless? That is, at least one of the mortgages defaults

Either none defect, or at least one does. The sum of the probabilities of these events is 100%. So

p + 47.22 = 100

p = 52.78

52.78% probability that the derivative from part (b) becomes worthless

or