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:
- 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.
- 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.
- 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.
- 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.