Question

Use Exercise 15 to show that \(P(E \cup F)=P(E)+P(F)-P(E F)\).

Short answer

In order to show that \(P(E \cup F) = P(E) + P(F) - P(EF)\), we used a Venn diagram to visualize the events E, F, and their union and intersection. The sum of probabilities of E and F contains the probability of their intersection twice, so we need to subtract it once to get the correct probability of the union: \(P(E \cup F) = P(E) + P(F) - P(EF)\). The derivation is based on the basic properties of probabilities and the definition of the union and intersection of events.

Step-by-step solution

Understand and define E, F, and their union and intersection

Before we start the proof, let's understand and define the events, their union, and intersection. - Event E: An event that occurs in a probability space. - Event F: Another event that occurs in the same probability space. - Union (\(E \cup F\)): This event occurs if either E or F or both occur. - Intersection (\(EF\)): This event occurs if both E and F occur together.

Recall the properties of probabilities

Now, let's recall some basic properties of probabilities: 1. \(0 \leq P(E) \leq 1\): Probability of an event always lies between 0 and 1. 2. \(P(E') = 1 - P(E)\): Probability of the complement of E is equal to one minus the probability of E. 3. \(P(A \cup B) = P(A) + P(B) - P(AB)\) for any two events A and B.

Derive the formula using Venn diagram and properties of probabilities

We will now use a Venn diagram to help us visualize and derive the formula for the probability of the union of two events. We can draw two overlapping circles, one for event E and the other for event F. There are three distinct regions in this diagram: - the region where only E occurs; - the region where only F occurs; - the region where both E and F occur (their intersection). The probability of the union of events E and F (the area covered by both circles) is given by the sum of the probabilities of each region: \(P(E \cup F) = P(E) + P(F) - P(EF)\). To show this, we can use the properties of probabilities mentioned in step 2: 1. Add the probabilities of E and F: \(P(E) + P(F)\). This sum includes the probability of their intersection twice because both E and F contain the region where they overlap. 2. Therefore, we have to subtract the intersection probability once: \(P(E) + P(F) - P(EF)\). 3. This gives us the final formula for the probability of the union of events E and F: \(P(E \cup F) = P(E) + P(F) - P(EF)\). As we have derived the formula using the Venn diagram and properties of probabilities, we have shown that \(P(E \cup F) = P(E) + P(F) - P(EF)\).