### Chapter 3. Probability: Probability

### Probability of the Difference

The probability of the intersection is not only used to calculate the probability of the union, but it is also used to calculate the probability of the difference. Recall that the difference of events #A# and #B# are all outcomes classified as #A#, but NOT as #B#: #A\backslash B#.

The difference of #A# and #B# thus contains all outcomes that are classified as #A#, minus the outcomes in #A# that are also classified as #B#.

So the difference of #A# and #B# is #A# minus the intersection of #A# and #B#:

\[\mathbb{P}(A\backslash B) = \mathbb{P}(A) − \mathbb{P}(A \cap B)\]

Consider the random experiment of rolling a die with six sides, numbered from #1# to #6#, and observing the number on top. For this experiment, we define the following events:

- #A =# 'The number is less than or equal to #3#'
- #B =# 'The number is odd'

The probabilities of events #A# and #B# are:

- #\mathbb{P}(A)=\cfrac{\text{number of outcomes}\leq 3}{\text{total number of outcomes}}=\cfrac{3}{6}#
- #\mathbb{P}(B) = \cfrac{\text{number of odd outcomes}}{\text{total number of outcomes}}=\cfrac{3}{6}#

In order to calculate the probability of the difference, we first need to calculate the probability of the intersection #\mathbb{P}(A \cap B)#. To do this, we need to know whether events #A# and #B# are independent.

If it is known that the outcome of the roll is a number #\leq 3#, then the probability of the roll being odd is #2# out of #3#; namely #1# and #3#, but not #2#:

\[\mathbb{P}(B|A) =\cfrac{2}{3}\]

This demonstrates that #\mathbb{P}(B) \neq \mathbb{P}(B|A)#, so we must conclude #A# and #B# are not independent. As such, the probability of the intersection is calculated as follows:

\[\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B|A) =\cfrac{3}{6}\cdot \cfrac{2}{3}=\cfrac{2}{6}\]

With this information, the probability of the difference of #A# and #B# can be calculated:

\[\mathbb{P}(A \backslash B) = \mathbb{P}(A) - \mathbb{P}(A \cap B) = \cfrac{3}{6}-\cfrac{2}{6}=\cfrac{1}{6}\]

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