If events A and B are mutually exclusive, and P(A) = x and P(B) = 2x, and P(A ∪ B) = 0.6, what is the value of P(B')?
Understand the Problem
The question states that events A and B are mutually exclusive, provides probabilities P(A) = x, P(B) = 2x, and P(A ∪ B) = 0.6. We need to find the value of P(B'). Since A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B). From this, we can find the value of 'x', then calculate P(B), and finally find P(B') using the fact that P(B') = 1 - P(B).
Answer
$P(B') = 0.6$
Answer for screen readers
$P(B') = 0.6$
Steps to Solve
- Use the mutually exclusive property
Since events A and B are mutually exclusive, we have:
$P(A \cup B) = P(A) + P(B)$
- Substitute given probabilities
Substitute the given probabilities $P(A) = x$, $P(B) = 2x$, and $P(A \cup B) = 0.6$ into the equation:
$0.6 = x + 2x$
- Solve for x
Combine like terms and solve for $x$:
$0.6 = 3x$
$x = \frac{0.6}{3} = 0.2$
- Calculate P(B)
Now that we have found $x$, we can calculate $P(B)$:
$P(B) = 2x = 2(0.2) = 0.4$
- Calculate P(B')
The probability of the complement of B, $P(B')$, is given by:
$P(B') = 1 - P(B)$
$P(B') = 1 - 0.4 = 0.6$
$P(B') = 0.6$
More Information
The probability of an event and its complement always adds up to 1, representing the entire sample space.
Tips
A common mistake is not using the property of mutually exclusive events correctly. For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$. For events that are not mutually exclusive, a different formula applies: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
Another common mistake is incorrectly calculating $P(B')$ by not subtracting $P(B)$ from 1.
AI-generated content may contain errors. Please verify critical information
