### Chapter 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.

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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:

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

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This probability of an event, based on the condition that another event has occurred is called a *conditional probability*.

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**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#

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?

For this random experiment, we define the following events:

- #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#

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

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