Calculating Mutually Exclusive Events Probability in a Sample Space
In probability theory, understanding the behavior of mutually exclusive events within a sample space is crucial. This article will explore the calculation of the probability of the union of two such events, A and C, given the probabilities of A, B, and C in a sample space S.
Given Information
We are provided with three events, A, B, and C, which are mutually exclusive and exhaustive in a sample space S. This means:
S A ∪ B ∪ C P(A) 1/5 * P(B) P(C) 4 * P(A) P(S) P(A) P(B) P(C) 1Step-by-Step Solution
Let's denote the probability of event A as P(A) x. Consequently, we can express:
P(B) 5x P(C) 4xGiven that the total probability of the sample space is 1, we can write:
[P(A) P(B) P(C) 1]
Substituting the expressions for P(B) and P(C), we get:
[x 5x 4x 1]
[1 1]
[x 1/10]
Therefore:
P(A) 1/10 P(B) 5/10 1/2 P(C) 4/10 2/5Finding P(A ∪ C)
Since A and C are also mutually exclusive, the probability of their union can be calculated as follows:
[P(A ∪ C) P(A) P(C)]
[ 1/10 2/5 1/10 4/10 5/10 1/2]
Hence, the final answer is:
[ boxed{1/2} ]
Alternatively, we can use the following property of mutually exclusive events:
[P(A ∪ C) 1 - P(B) 1 - 1/2 1/2]
Conclusion
The probability that either event A or C occurs, given the conditions provided, is 1/2. This calculation is a fundamental application of probability theory and demonstrates the importance of understanding mutually exclusive and exhaustive events within a sample space.