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Two events, [tex]E_1[/tex] and [tex]E_2[/tex], are defined for a random experiment. What is the probability that at least one of the two events occurs in any trial of the experiment?

A. [tex]P\left(E_1\right) - P\left(E_2\right) - P\left(E_1 \cap E_2\right)[/tex]

B. [tex]P\left(E_1\right) + P\left(E_2\right) + P\left(E_1 \cap E_2\right)[/tex]

C. [tex]P\left(E_1\right) + P\left(E_2\right) - 2 P\left(E_1 \cap E_2\right)[/tex]

D. [tex]P\left(E_1\right) + P\left(E_2\right) - P\left(E_1 \cap E_2\right)[/tex]

Sagot :

GinnyAnswer
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]
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