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In these lessons, we will learn how to find the probability of dependent events. We will also learn the difference between the probability of dependent events and the probability of independent events.

Related Topics: More Lessons on Probability

Events are dependent if the outcome of one event affects the outcome of another. For example, if you draw two colored balls from a bag and the first ball is not replaced before you draw the second ball then the outcome of the second draw will be affected by the outcome of the first draw.

If *A* and *B* are dependent events, then the probability of *A* happening **AND** the probability of *B* happening, given *A, *is P(*A*) **×** P(*B *after *A*).

P(*A* and *B*) = P(*A*) **× **P(*B* after *A*)

P(*B* after *A*) can also be written as P(*B* | *A*)

then P(*A* and *B*) = P(*A*) **× **P(*B* | *A*)

* Example: *

A purse contains four $5 bills, five $10 bills and three $20 bills. Two bills are selected without the first selection being replaced. Find P($5, then $5)

* Solution: *

There are four $5 bills.

There are a total of twelve bills.

P($5) =

The result of the first draw affected the probability of the second draw.

There are three $5 bills left.

There are a total of eleven bills left.

P($5 after $5) =

P($5, then $5) = P($5) · P($5 after $5) =

The probability of drawing a $5 bill and then a $5 bill is

* Example: *

A bag contains 6 red, 5 blue and 4 yellow marbles. Two marbles are drawn, but the first marble drawn is not replaced.

a) Find P(red, then blue)

b) Find P(blue, then blue)

* Solution: *

a) There are 6 red marbles.

There are a total of 15 marbles.

P(red) =

The result of the first draw affected the probability of the second draw.

There are 5 blue marbles.

There are a total of 14 marbles left.

P(blue after red) =

P(red, then blue) = P(red) · P(blue after red) =

The probability of drawing a red marble and then a blue marble is

b) There are 5 blue marbles.

There are a total of 15 marbles.

P(blue) =

The result of the first draw affected the probability of the second draw.

There are 4 blue marbles left.

There are a total of 14 marbles left.

P(blue after blue) =

P(blue, then blue) = P(blue) · P(blue after blue) =

The probability of drawing a red marble and then a blue marble is

The following videos show more examples of calculating the probability of dependent events.

We have a box with 10 red marbles and 10 blue marbles. Find P(drawing two blue marbles).

Probability of Dependent Events

A club of 9 people wants to choose a board of 3 officers: President, Vice-President and Secretary. Assuming that the officers are chosen at random, what is the probability that the officers are Marsha for President, Sabita for Vice-President and Robert for Secretary?

Probability Dependent Events

How do you calculate probability of two dependent events?

A box contains 3 pens, 2 markers and 1 highlighter. Tara selects one item at random and does not return it to the box. She then selects a second item at random. What is the probability that Tara selects one pen and then one marker?

Andrea has 8 blue socks and 4 red socks in her drawer. She chooses one sock at random and puts it on. She then chooses another sock without looking. Find the probability of the following event P(red, then red).

Statistics - Dependent and Independent Events

This lesson teaches the distinction between Independent and Dependent Events, and how to calculate the probability of each.

You can use the Mathway widget below to practice Statistics or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.