Conditional Probability

3. Probability: Probability

Conditional Probability

For experiments that observe multiple events, it might be possible that the outcome of one event influences the outcome of another event.
$\phantom{0}$

Outcomes influencing each other

For example, when rolling two dice, you might be interested in the probability that the total score of the two dice equals $2$ . There is only one way to obtain a total score of $2$ , which is to roll a $1$ with both dice. The probability of this event is:

$\mathbb{P}(\text{'the total score is 2'}) = \cfrac{1}{36}$

Now, suppose the dice are thrown one at a time and the first die comes up $1$ . This directly influences the probability of obtaining a total score of $2$ . In order to get a total score of $2$ , given that the first die came up $1$ , you need to roll a $1$ with the second die as well. The probability of this happening is:

$\mathbb{P}(\text{'the total score is 2, given that the first die is a 1'}) = \mathbb{P}(\text{'the second die comes up 1'}) = \cfrac{1}{6}$

$\phantom{0}$
This probability of an event, based on the condition that another event has occurred is called a conditional probability.
$\phantom{0}$

Conditional Probability

Definition

A conditional probability is the probability of an event A occurring, given that an event B has occurred.

Notation

$\mathbb{P}(A|B)$

Rules

The probability of $A$ given $B$ equals the probability of $A$ AND $B$ , divided by the probability of $B$ :

$\mathbb{P}(A|B)=\cfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$
Once event $A$ occurs, it is certain that event $A$ occurs:

$\mathbb{P}(A|A)=1$
IF $A$ and $B$ are mutually exclusive, then:

$\mathbb{P}(A|B)=0$

A total of $100$ students, $40$ males and $60$ females, are surveyed about their handedness. The results of the survey reveal that among these $100$ students there are a total of $12$ left-handed students, $4$ males and $8$ females.

Out of these $100$ students, a single student is selected at random. What is the probability that this student is left-handed, given that the student is female?

$\mathbb{P}(A|B)=0.133$

Events $A$ and $B$ are defined as follows:

$A =$ 'the student is left-handed'
$B =$ 'the student is female'

The probability of randomly selecting a left-handed student, given that the student is female corresponds to the following conditional probability:

$\mathbb{P}(A|B) = \dfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$

The probabilities needed for this calculation are:

$\mathbb{P}(A \cap B) = \dfrac{\text{number of left-handed female students}}{\text{total number of students}}=\dfrac{8}{100}=0.08$
$\mathbb{P}(B) = \dfrac{\text{number of female students}}{\text{total number of students}} = \dfrac{60}{100}=0.60$

Entering these probabilities into the equation gives:

$\mathbb{P}(A|B)=\cfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}=\dfrac{0.08}{0.60}=0.133$

New example