Probability: Finding the largest and smallest value of two events

probability

The question states: Let A, B (subset of sample space) be events. Given P(A)= 0.6 and P(B) = 0.5.

  1. What is the largest and smallest that P(A∩B) can take?

  2. What is the largest and smallest that P(A∪B) can take?

I only know that for P(A∩B), the smallest value is acquired by P(A) + P(B) – P(A∩B) = 1, where P(A∩B) = 0.6 + 0.5 – 1 = 0.1. Finding the largest value however, is not quite intuitive for me so can anyone please explain how I'm supposed to solve these type of problems?

Best Answer

Note that $P(A\cap B) + P(A\cup B) = P(A) + P(B) = 1.1$ is a fixed number, so whatever case minimizes $P(A\cap B)$ will maximise $P(A\cup B)$, and vice versa.

You've shown that the minimal value of $P(A\cap B)$ is $0.1$. What maximal value of $P(A\cup B)$ does this correspond to?

As for the maximal value of $P(A\cap B)$, note that $A\cap B\subseteq A, B$, so $P(A\cap B)\leq P(A), P(B)$. What bound on the value of $P(A\cap B)$ does this give? Is it possible to reach this bound? What does this give for the minimal value of $P(A\cup B)$?

please explain how I'm supposed to solve these type of problems?

This is very difficult to give a good answer to. "Practice and get experience" is the only general tip I know. As for concrete techniques, it's mostly a matter of just listing things that you know about the problem at hand (either mentally, or on a piece of paper), and what you think you need to figure out. Once those two lists are bridged, you have a solution.

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