Question
The probability for event A is 0.3, the probability for event B is 0.6, and the probability of events A or B is 0.8.
Why are the events not mutually exclusive?
The sum of P(A) and P(B) is less than P(A or B).
The product of P(A) and P(B) is less than P(A or B).
The product of P(A) and P(B) is not equal to P(A or B).
The sum of P(A) and P(B) is not equal to P(A or B).
Why are the events not mutually exclusive?
The sum of P(A) and P(B) is less than P(A or B).
The product of P(A) and P(B) is less than P(A or B).
The product of P(A) and P(B) is not equal to P(A or B).
The sum of P(A) and P(B) is not equal to P(A or B).
Answers
Answered by
GPT-4o mini
To determine why events A and B are not mutually exclusive, let's first clarify what it means for events to be mutually exclusive. Mutually exclusive events cannot occur at the same time, meaning the probability of both events happening together is zero.
If A and B were mutually exclusive, the probability of A or B (denoted as \( P(A \text{ or } B) \)) would be calculated as:
\[
P(A \text{ or } B) = P(A) + P(B)
\]
Substituting in the probabilities given:
\[
P(A \text{ or } B) = 0.3 + 0.6 = 0.9
\]
Since we have that \( P(A \text{ or } B) \) is actually given as 0.8, it indicates that there is overlap between events A and B. To have \( P(A \text{ or } B) \) equal to 0.8 while the sum of their individual probabilities equals 0.9 implies that their intersection (when both events occur simultaneously) is non-zero.
Using the formula for the probability of A or B:
\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\]
We can rearrange this to find \( P(A \text{ and } B) \):
\[
P(A \text{ and } B) = P(A) + P(B) - P(A \text{ or } B)
\]
Substituting our values:
\[
P(A \text{ and } B) = 0.3 + 0.6 - 0.8 = 0.1
\]
Since \( P(A \text{ and } B) = 0.1 \) is greater than 0, this confirms that A and B can occur together, thus they are not mutually exclusive.
Now, evaluating your provided options:
1. The sum of P(A) and P(B) is less than P(A or B). **(False)**
2. The product of P(A) and P(B) is less than P(A or B). **(Not applicable here)**
3. The product of P(A) and P(B) is not equal to P(A or B). **(Not applicable here)**
4. The sum of P(A) and P(B) is not equal to P(A or B). **(True)**
Therefore, the correct statement that explains why events A and B are not mutually exclusive is:
**The sum of P(A) and P(B) is not equal to P(A or B).**
If A and B were mutually exclusive, the probability of A or B (denoted as \( P(A \text{ or } B) \)) would be calculated as:
\[
P(A \text{ or } B) = P(A) + P(B)
\]
Substituting in the probabilities given:
\[
P(A \text{ or } B) = 0.3 + 0.6 = 0.9
\]
Since we have that \( P(A \text{ or } B) \) is actually given as 0.8, it indicates that there is overlap between events A and B. To have \( P(A \text{ or } B) \) equal to 0.8 while the sum of their individual probabilities equals 0.9 implies that their intersection (when both events occur simultaneously) is non-zero.
Using the formula for the probability of A or B:
\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\]
We can rearrange this to find \( P(A \text{ and } B) \):
\[
P(A \text{ and } B) = P(A) + P(B) - P(A \text{ or } B)
\]
Substituting our values:
\[
P(A \text{ and } B) = 0.3 + 0.6 - 0.8 = 0.1
\]
Since \( P(A \text{ and } B) = 0.1 \) is greater than 0, this confirms that A and B can occur together, thus they are not mutually exclusive.
Now, evaluating your provided options:
1. The sum of P(A) and P(B) is less than P(A or B). **(False)**
2. The product of P(A) and P(B) is less than P(A or B). **(Not applicable here)**
3. The product of P(A) and P(B) is not equal to P(A or B). **(Not applicable here)**
4. The sum of P(A) and P(B) is not equal to P(A or B). **(True)**
Therefore, the correct statement that explains why events A and B are not mutually exclusive is:
**The sum of P(A) and P(B) is not equal to P(A or B).**