To determine the probability that at least one of the two events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] occurs, we use the principle of inclusion-exclusion for probabilities.
According to this principle, the probability that at least one of the two events occurs, denoted as [tex]\( P(E_1 \cup E_2) \)[/tex], is given by:
[tex]\[ P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2) \][/tex]
Here is a step-by-step explanation:
1. Identify Individual Probabilities: First, determine the probabilities of the individual events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex]. Let's denote [tex]\( P(E_1) \)[/tex] and [tex]\( P(E_2) \)[/tex] as the probabilities of events [tex]\( E_1 \)[/tex] and [tex]\( E_2 \)[/tex] occurring, respectively.
2. Understand Intersection Probability: Next, consider the probability that both events occur simultaneously, denoted as [tex]\( P(E_1 \cap E_2) \)[/tex].
3. Apply the Inclusion-Exclusion Principle:
- The probability that either event [tex]\( E_1 \)[/tex] or event [tex]\( E_2 \)[/tex] happens (or both) can be calculated by summing the probabilities of each event and then subtracting the probability that both events occur simultaneously. This avoids double-counting the overlap between the two events.
So, the formula to compute the probability that at least one of the two events occurs is:
[tex]\[ P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2) \][/tex]
Given this formula, the correct answer to the question is:
D. [tex]\( P(E_1) + P(E_2) - P(E_1 \cap E_2) \)[/tex]
