Question

a. Suppose events \(E\) and \(F\) are mutually exclusive with \(P(E)=0.41\) and \(P(E)=0.23\). i. What is the value of \(P(E \cap F) ?\) ii. What is the value of \(P(E \cup F) ?\) b. Suppose that for events \(A\) and \(B, P(A)=0.26, P(B)=0.34\), and \(P(A \cup B)=0.47 .\) Are \(A\) and \(B\) mutually exclusive? How can you tell?

Short answer

i. \(P(E \cap F) = 0\), ii. \(P(E \cup F) = 0.64\); Events \(A\) and \(B\) are not mutually exclusive

Step-by-step solution

Find \(P(E \cap F)\) for mutually exclusive events

From the definition of mutually exclusive events, we know that two events cannot occur simultaneously. Thus, \(P(E \cap F) = 0\)

Find \(P(E \cup F)\) for mutually exclusive events

The union of two mutually exclusive events is found by adding their individual probabilities. Thus, \(P(E \cup F) = P(E) + P(F) = 0.41 + 0.23 = 0.64\)

Determine if events \(A\) and \(B\) are mutually exclusive

We know that if events are mutually exclusive then \(P(A \cup B) = P(A) + P(B)\). Let's compare \(0.26 + 0.34\) with \(0.47\). Since \(0.26 + 0.34 = 0.6\), which is not equal to \(0.47\), therefore events \(A\) and \(B\) are not mutually exclusive.

Key concepts

Probability of Union of Events

Understanding the probability of the union of events involves evaluating the likelihood of either one event, another event, or both events occurring at the same time. Mathematically, for any two events A and B, this is represented as \(P(A \cup B)\), which reads as the probability of A or B occurring.

Now, to calculate \(P(A \cup B)\), we generally add the probabilities of A and B and subtract the probability of their intersection, which is \(P(A \cap B)\). This subtraction is necessary because if we simply added the probabilities of A and B without considering the intersection, we would be counting the intersection twice. However, there's an exception to this rule when dealing with mutually exclusive events, which we'll explore in other sections.

Probability of Intersection of Events

The probability of the intersection of events refers to the likelihood of two or more events occurring simultaneously. Represented as \(P(A \cap B)\), it illustrates the intersection of event A with event B.

However, to determine this intersection, we need to consider if the events can actually occur at the same time. For independent events, where one event's occurrence doesn't affect the other's, we can find the intersection by multiplying the probabilities of the individual events (i.e., \(P(A) \times P(B)\)). But the scenario is different for mutually exclusive events; their intersection probability is always zero because they cannot occur simultaneously.

Definition of Mutually Exclusive Events

Mutually exclusive events are cases in which the occurrence of one event completely prevents the occurrence of another. In other words, these events cannot happen at the same time.

For example, when you flip a coin, the event of landing tails is mutually exclusive from the event of landing heads. As a result, if two events E and F are mutually exclusive, then by definition, \(P(E \cap F) = 0\), meaning the probability of their intersection is zero. This also simplifies the calculation of their union, since \(P(E \cup F) = P(E) + P(F)\), with no need to subtract anything for their intersection because it's non-existent.

Probability Principles

There are core principles in probability that provide the foundation for understanding the behavior of random phenomena. One fundamental principle is the addition rule, which helps in determining the probability of the union of two events, considering if they are mutually exclusive or not.

The multiplication rule is another key principle, used to calculate the probability of the intersection of independent events. Moreover, it's important to distinguish between independent and mutually exclusive events, as this affects how we apply these principles: for mutually exclusive events, there is no overlap, while independent events can happen at the same time without influencing each other. Understanding and correctly applying these principles is essential for accurate probability calculations.