Question

Suppose \(P(A)=.1\) and \(P(B)=.5 .\) $$\text { If } P(A \cup B)=.65, \text { are } A \text { and } B \text { mutually }$$$$\text { exclusive? }$$

Short answer

Answer: No, events A and B are not mutually exclusive.

Step-by-step solution

Write down given probabilities

We begin by listing the probabilities provided to us in the problem: \(P(A) = 0.1\) \(P(B) = 0.5\) \(P(A \cup B) = 0.65\)

Calculate combined probability assuming A and B are mutually exclusive

Since mutually exclusive events have \(P(A ∩ B) = 0\), we first assume A and B are mutually exclusive and calculate the combined probability using the formula \(P(A ∪ B) = P(A) + P(B)\): \(P(A ∪ B) = 0.1 + 0.5 = 0.6\)

Compare calculated combined probability with the given probability

Now we will compare our calculated combined probability of 0.6 with the given probability of the union: \(0.6 \neq 0.65\) Since the calculated combined probability assuming mutually exclusive events (0.6) does not match the given probability of the union (0.65), we can conclude:

Conclusion

Events A and B are not mutually exclusive since our calculated probability does not match the given probability of their union.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with quantifying the likelihood of various events occurring. Think of it as the mathematical framework for understanding and measuring uncertainty. In essence, when we speak about the probability of an event, we're referring to a number between 0 and 1, where 0 means the event will never occur, and 1 guarantees that the event will happen. For example, flipping a fair coin gives us a probability of 0.5 for landing on heads, because the event is as likely to occur as it is not to occur.

In practical applications, probability can be used to assess risk, make predictions, and guide decision-making processes. The fundamental principle of probability theory is that the probabilities of all possible outcomes must sum up to one, ensuring that the likelihood of 'something' occurring is absolute.

Union of Events

The union of events in probability refers to the event that at least one of several events will occur. Mathematically, the union of two events, A and B, is denoted as \(A \cup B\) and includes all outcomes that are either in A, or B, or in both. To calculate the probability of the union of two events, we generally add the probability of each event, but we must adjust for any overlap because the overlapping outcomes are included in the calculations for both A and B.

For mutually exclusive events — events that cannot happen at the same time — the probability of their union simplifies to just the sum of their probabilities, because there is no overlap. This is not present in non-mutually exclusive events where the intersection must be accounted for. For example, if a dice roll could result in either an even number (A) or a number greater than two (B), the events are not mutually exclusive because the outcomes '4' and '6' fit both criteria.

Intersection of Events

The intersection of events represents the set of outcomes that are common to all events under consideration. If we consider events A and B, their intersection, denoted by \(A \cap B\), includes all outcomes that are both in A and in B. The probability of the intersection of two events is calculated as \(P(A \cap B)\).

In the context of mutually exclusive events, the intersection is an empty set because mutually exclusive events cannot occur simultaneously, hence \(P(A \cap B) = 0\). However, if events are not mutually exclusive, there could be a positive probability that they will occur together, and thus their intersection is not empty. This was the crux behind the exercise posed, where the assumption of A and B being mutually exclusive led to an incorrect calculation of the union's probability, as it did not account for their actual intersection.