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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(E_1) - P(E_2) - P(E_1 \cap E_2)[/tex]

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

C. [tex]P(E_1) + P(E_2) + P(E_1 \cap E_2)[/tex]

D. [tex]P(E_1) + P(E_2) - 2 P(E_1 \cap E_2)[/tex]

Sagot :

GinnyAnswer
To find the probability that at least one of the two events [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] occurs in any trial of the experiment, we need to use the formula for the union of two probabilities. The correct formula for the probability that at least one of two events occurs is given by:

[tex]\[ P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2) \][/tex]

This formula accounts for the fact that when you add the probabilities of [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex], the intersection (where both events happen) is counted twice. Therefore, you subtract the intersection once to get the correct probability of at least one event occurring.

Given this information, we can determine that the correct answer is:

[tex]\[ \boxed{B. \ P(E_1) + P(E_2) - P(E_1 \cap E_2)} \][/tex]
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