Chapter 3

Section 3.6

Conditional Probability

Conditional Probability

The interesting questions that probability can answer are how much one event will effect another.

Does attending class effect your chance of passing?Does a diamond effect your chance of winning a poker hand?Does a taking a drug increase your chance of developing a tumor?

In mathematics we are not just interested in saying that one thing influences another but by measuring by how much (i.e. A numerical amount).

The tool that mathematics uses to measure this influence of one event on another is called conditional probability.

This is its own type of event that is made up of two other events. It is similar to what we have done before when two events are associated with the same experiment.

P(A∩B) = The chance both A and B occur at the same time.

P(AB) = The chance either A or B occur.

P(A|B) = The chance that A can still happen if you assume or know that B has occurred. This is read "the probability of A given B". We call this the conditional probability.

Formula for Conditional Probability

The conditional probability of two events A and B denoted P(A|B) can be computed by calculating the chance A can happen if B has happened. This means that B has become the new sample space. For A to still happen means the event must be in A∩B.

SA B

)(

)(

)(

)()|(

BP

BAP

Binnumber

BAinnumber

Bn

BAnBAP

This step is a technical algebra detail.

Example:

Experiment: Flip 3 coinsS={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}M: The event the majority are heads.F : The event the first coin is a head.

P(M) =The chance the majority are heads =

S

M F

HHHHHTHTH

THH HTT THT

TTHTTT

P(M|F) = The chance the majority are heads given or assuming the first is a head.

%508

4

%754

3

Having the first be a head increases the chance of the majority being a head by 25%!

Example

A researcher has 50 mice. She gives 25 of the mice a new drug for three months and the others no drug at all. After 3 months she finds that 16 mice have developed tumors. Of the 16 mice with tumors 6 where given the drug.

An experiment is conducted where you pick a mouse at random from this group. We are interested in the following two events:

T: The mouse develops a tumor.

D: The mouse was given a drug.

P(T) = The chance a mouse develops a tumor =

P(T|D) = The chance a mouse develops a tumor if they take the drug=

Does taking the drug increase, decrease or keep the same a mouse's chance of developing a tumor?

If increase or decrease by how much?

S

T D

610 1915

%3250

16

%2425

6

Decreases

8% (i.e. 32%-24%)

Example

The table to the right shows the breakdown of 1000 students at a small college who are in or not in a greek club and who have made or not made the Dean's List.

Dean's List Not on Dean's List

Greek Club 90 210

Not in a Greek Club

250 450

If a student is chosen at random what is the chance they will be on the Dean's List?

%341000

340')'(

schoolinnumber

ListsDeanonnumberListsDeanP

If a student is chosen at random what is the chance they will be on the Dean's List if you assume they are in a greek club?

%30300

90

lublub)|'(

CGreekinnumber

bothinnumberCGreekListsDeanP

Does being in a Greek Club increase, decrease or keep the same a students chance of being on the Dean's List? If increase or decrease by how much?

Decreases by 4%

Conditional Probability and Probability Trees

The second formula for conditional probability can be rewritten as shown to the right. This means that you can find the chance both events A and B occur by multiplying by the conditional probability.

)()|()(

)(

)()|(

BPBAPBAP

BP

BAPBAP

This helps to organize the probability information in the form of a tree.

Example:

A bag contains 3 red (R) and 2 blue (B) balls. Two balls are picked out without replacement. What is the probability of picking each of the various color combinations of balls?

R

B

R

R

B

B

P(RR) =

P(RB) =

P(BR) =

P(BB) =

53

52

42

42

43

41

%3020

6

4

2

5

3

%3020

6

4

2

5

3

%3020

6

4

3

5

2

%1020

2

4

1

5

2

* It is important when you fill in the probabilities in the tree itself you use fractions or decimals, not percentages.