Question Video: Determining the Probability of the Intersection of Two Independent Events | Nagwa Question Video: Determining the Probability of the Intersection of Two Independent Events | Nagwa

# Question Video: Determining the Probability of the Intersection of Two Independent Events Mathematics

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?

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### 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.

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