3. Probability: Probability
Independence
Conditional probabilities concern two events that influence one another. This is not always the case, however, as some events do not influence the outcomes of other events at all. For example, when rolling two dice, rolling a 6 with the first die does not influence the probability of rolling a 6 with the second one. Such events are said to be independent of each other.
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Independence
Definition
Two events #A# and #B# are independent of each other if the occurrence of one event has no influence on the probability of the other event occurring.
Examples
- Whether it will rain today is independent of whether you roll a 1 on a dice.
- Whether you will pass this course is independent of whether you get heads on a coin toss.
Rules
- If #A# is independent of #B#, then #B# is also independent of #A#.
- If #A# and #B# are independent, then:
#\mathbb{P}(A) = \mathbb{P}(A|B)# or equivalently, #\mathbb{P}(B) = \mathbb{P}(B|A)# - If #A# and #B# are independent, then
#\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)#
Verify the independence of two dice rolls.
Define the following events:
- #A=# 'the first die lands on #1#'
- #B=# 'the second die lands on #1#'
- #A \cap B=# 'both dice land on #1#'
The probabilites corresponding to these events are:
- #\mathbb{P}(A) = \cfrac{1}{6}#
- #\mathbb{P}(B) = \cfrac{1}{6}#
- #\mathbb{P}(A \cap B) = \cfrac{1}{36}#
Using this information, calculate the probability of #A# given #B#:
\[\mathbb{P}(A|B) = \cfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}=\dfrac{\cfrac{1}{36}}{\cfrac{1}{6}}=\cfrac{1}{6}\]
It is known that #\mathbb{P}(A) =\cfrac{1}{6}#, so this means that #\mathbb{P}(A) = \mathbb{P}(A|B)#. Therefore, according to the rules, #A# and #B# are independent.
