### Video Transcript

Suppose 𝐴 and 𝐵 are two events
with probability 𝑃 of 𝐴 equals 0.6 and 𝑃 of 𝐵 equals 0.5. Given that the probability of 𝐴
intersection 𝐵 is 0.4, what is the probability that neither of the events
occur?

In this question, we need to
calculate the probability that neither of the events occur. As shown on the Venn diagram, this
is the same as the complement of the union of events 𝐴 and 𝐵. As an event and its complement sum
to one, we can therefore calculate the probability that neither event 𝐴 nor event
𝐵 occur by subtracting the probability of 𝐴 union 𝐵 from one. In this question, we are not given
the probability of the union. However, we are given the
probability of event 𝐴, the probability of event 𝐵, and the probability of the
intersection of events 𝐴 and 𝐵. This means that we can begin by
using the additive rule of probability, which states that the probability of 𝐴
union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the
probability of 𝐴 intersection 𝐵.

Substituting in the values given,
the right-hand side becomes 0.6 plus 0.5 minus 0.4. The probability of 𝐴 union 𝐵 is
equal to 0.7. This is the area shaded in pink on
our Venn diagram. We can now find a complement of
this by subtracting 0.7 from one, which is equal to 0.3. We can therefore conclude that if
the probability of 𝐴 is 0.6, the probability of 𝐵 is 0.5, and the probability of
𝐴 intersection 𝐵 is 0.4, then the probability that neither event 𝐴 nor event 𝐵
occurs is 0.3. This is the section outside of the
circles on our Venn diagram.