The probability of the union of two events with nonzero probabilities

The probability of the union of two events with nonzero probabilities

A

cannot be less than one.

B

cannot be one.

C

cannot be less than one and cannot be one.

D

None of the other answers is correct.

The Correct Answer and Explanation is :

The correct answer is D) None of the other answers is correct.

Explanation:

Let’s denote two events as ( A ) and ( B ). The probability of the union of two events, ( P(A \cup B) ), is the probability that at least one of the events ( A ) or ( B ) occurs. It is given by the formula:

[
P(A \cup B) = P(A) + P(B) – P(A \cap B)
]

This formula accounts for the fact that the probability of both events happening simultaneously (i.e., the intersection ( A \cap B )) is subtracted to avoid double-counting.

Now, let’s analyze each of the given options:

1. A) Cannot be less than one.

• This is incorrect. While the probability of an event is always between 0 and 1, the union of two events can never exceed a probability of 1. For example, if ( P(A) = 0.8 ) and ( P(B) = 0.7 ), the probability of ( P(A \cup B) ) can be at most 1, but it is typically less than 1 unless the events are perfectly correlated.

1. B) Cannot be one.

• This is also incorrect. It is possible for the union of two events to have a probability of 1. For instance, if ( A ) and ( B ) cover the entire sample space (i.e., one or both will always occur), the union ( A \cup B ) will have a probability of 1.

1. C) Cannot be less than one and cannot be one.

• This is incorrect as well. The union of two events can indeed be less than 1 or equal to 1, depending on the relationship between the events and their probabilities.

1. D) None of the other answers is correct.

• This is the correct answer. The probability of the union of two events can range from 0 to 1, inclusive, depending on the overlap between the events. There is no restriction that prevents it from being less than 1 or equal to 1.

Summary:

The probability of the union of two events ( A \cup B ) can indeed be any value between 0 and 1, depending on the individual probabilities of ( A ) and ( B ), and their intersection. Therefore, none of the other provided options is entirely correct.