### Video Transcript

David and Olivia applied for life
insurance. The company has estimated that the
probability that David will live to be at least 85 years old is 0.6. And the probability that Olivia
will live to be at least 85 years old is 0.25. Given that these are independent
events, what is the probability they will both live to be at least 85?

Two events are said to be
independent if the outcome of one does not affect the outcome of the other. We know that if two events are
independent, then the probability of A and B or A intersection B is equal to the
probability of A multiplied by the probability of B. If we let event A be the
probability that David lives to at least 85, then the probability of A is 0.6. If event B is the probability that
Olivia lives to 85 or more, then the probability of B is 0.25. As these events are independent, we
can calculate the probability of both by multiplying 0.6 by 0.25. This is equal to 0.15. The probability that both David and
Olivia live to be at least 85 is 0.15.

We could show this information on a
Venn diagram. The overlap in the two circles A
and B is the probability of both events occurring. And this is equal to 0.15. We know that the probability of A
was 0.6. As 0.6 minus 0.15 is 0.45, the
probability of only A occurring is 0.45. Likewise, the probability of only
event B occurring is 0.1 as 0.25 minus 0.15 is equal to 0.1. We know that probabilities must sum
to one. Therefore, there must be 0.3
outside of our circles. This is because the sum of 0.45,
0.15, and 0.1 is 0.7. And one minus this is equal to
0.3. This 0.3 represents the probability
that neither David nor Olivia live to be 85.