The law of total probability

Theorem:

Let B be an event with and . Then for any event A,

.

Example:

An urn contains 10 white and 6 red balls. Two balls are selected at random without replacement. What is the probability that second selected ball is red?

Solution:

Let A be the event that second selected ball is red, B be event that the first ball is white. Then , , , . Then by the law of total probability:

= .

Theorem:

Let be a set of nonempty, mutually exclusive subsets of the sample space S and for then for any event

A of S,

.

Example:

Suppose that 70% of seniors, 60% of juniors, 55% of the sophomores, and 40% of the freshmen of a university use the library frequently. If 35% of all students are freshmen, 30% are sophomores, 20% are juniors, and 15% are seniors, what percent of all students use the library frequently?

Solution:

Let A be the event that a randomly selected student is using library frequently. Let F, O, J, and E be the events that he or she is a freshmen, sophomore, junior, or senior respectively. Thus

.

Therefore, 53% of these students use the library frequently.

Дата добавления: 2015-08-05