### Video Transcript

Denote by 𝐴 and 𝐵 two events with
probabilities the probability of 𝐴 equals 0.58 and the probability of 𝐵 equals
0.2. Given that the probability of 𝐴
union 𝐵 equals 0.64, find the probability of 𝐴 intersects 𝐵.

In this question, we are given the
probabilities of two events 𝐴 and 𝐵, as well as the probability of the union of
these events, that is, the probability of either event occurring. We want to use these probabilities
to find the probability of the intersection of these events, that is, the
probability of both events occurring.

We can do this by noting that we
are given three of the four probabilities in the addition rule for probability and
asked to find the fourth probability. We can recall that the addition
rule for probability tells us that for any events 𝐴 and 𝐵, the probability of
either event occurring, 𝑃 of 𝐴 union 𝐵, is equal to the probability of 𝐴
occurring added to the probability of 𝐵 occurring minus the probability of both
events occurring, that is the probability of 𝐴 intersects 𝐵.

We can substitute the given
probabilities into the addition rule for probability to obtain 0.64 is equal to 0.58
plus 0.2 minus the probability of 𝐴 intersects 𝐵. We can then rearrange this equation
to solve for the unknown probability. We have 𝑃 of 𝐴 intersects 𝐵 is
equal to 0.58 plus 0.2 minus 0.64. We can then evaluate this
expression to obtain that the probability of both events 𝐴 and 𝐵 occurring is
0.14.