### Chapter 3. Probability: Probability

### Probability of the Intersection

Quite often, it is needed to calculate the probability of the intersection of two events #A# and #B#. Recall that the intersection #A# and #B# is the set of outcomes that are classified as #A# AND #B#.

To calculate the probability of the intersection of two events, apply the following rules:

Rules

*Multiplication rule*: The probability that events #A# and #B# both occur is equal to the probability that event #A# occurs multiplied by the probability that event #B# occurs, given that #A# has occurred:

#\mathbb{P}(A\cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B|A)#- If #A# and #B# are independent, then

#\mathbb{P}(A\cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)#

Consider the random experiment of rolling two dice with six sides, numbered from #1# to #6#, and observing the numbers on top.

If we define event #A# as 'at least one number is a #6#', what is the probability of #A# occurring?

One way to calculate the probability of the event 'at least one number is a #6#' is to make use of the complement rule:

\[\mathbb{P}(A) = 1 − \mathbb{P}(A^c)\]

Here, #A^c# is the event of getting no sixes in two rolls.

To calculate the value of #\mathbb{P}(A^c)#, we define events #B_1# and #B_2# as follows:

- #B_1=# 'the first roll is not a #6#'
- #B_2=# 'the second roll is not a #6#'

The probabilities of these events are:

- #\mathbb{P}(B_1) = \cfrac{5}{6}#
- #\mathbb{P}(B_2) = \cfrac{5}{6}#

The event of neither roll being a #6# is equivalent to the combined event of getting no #6# on the first roll AND getting no #6# on the second roll:

\[A^c = B_1 \cap B_2\]

Since dice rolls are independent events, we can use the following formula to calculate the probability of the intersection of #B_1# and #B_2#:

\[\mathbb{P}(A^c)=\mathbb{P}(B_1 \cap B_2) = \mathbb{P}(B_1) \cdot \mathbb{P}(B_2) = \frac{5}{6}\cdot \frac{5}{6}=\frac{25}{36}\]

Now that #\mathbb{P}(A^c)# is known, #\mathbb{P}(A)# can be calculated:

\[\mathbb{P}(A) = 1 − \mathbb{P}(A^c) =\frac{11}{36}\]

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