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