What is an event in probability that is not mutually exclusive? Can you provide one or two samples?
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Two events that are NOT mutually exclusive are events that "overlap," that is, if one happens the other one may still happen.
Examples
1. x > 4 and x < 5 are not mutually exclusive. If x = 4.5 then both are true.
2. x = 2, 3 and x = 1, 2 are not mutually exclusive. If x = 2 then both are true. However if x = 1, 3 then only one is true. So if one is true the other may or may not happen.
In set notation, A ∩ B ≠ Ø and P(A ∪ B) = P(A) + P(B) - P(A ∩ B). The reason is that the intersection is not empty. So you add the probability of set A and the probability of set B, but now you double counted the intersection, so you subtract the probability of the intersection.
Examples:
1. Roll a 6 sided die. P(X = 1, 2, 3) = 1/2. P(X = 1, 2, 6) = 1/2. But the intersection is X = 1, 2 and P(X = 1, 2) = 1/3, so P(X = 1, 2, 3, and X = 1, 2, 6) = 1/2 + 1/2 - 1/3 = 2/3. Note the union is X = 1, 2, 3, 6 so P(X = 1, 2, 3, 6) = 2/3 which is the same.
Examples
1. x > 4 and x < 5 are not mutually exclusive. If x = 4.5 then both are true.
2. x = 2, 3 and x = 1, 2 are not mutually exclusive. If x = 2 then both are true. However if x = 1, 3 then only one is true. So if one is true the other may or may not happen.
In set notation, A ∩ B ≠ Ø and P(A ∪ B) = P(A) + P(B) - P(A ∩ B). The reason is that the intersection is not empty. So you add the probability of set A and the probability of set B, but now you double counted the intersection, so you subtract the probability of the intersection.
Examples:
1. Roll a 6 sided die. P(X = 1, 2, 3) = 1/2. P(X = 1, 2, 6) = 1/2. But the intersection is X = 1, 2 and P(X = 1, 2) = 1/3, so P(X = 1, 2, 3, and X = 1, 2, 6) = 1/2 + 1/2 - 1/3 = 2/3. Note the union is X = 1, 2, 3, 6 so P(X = 1, 2, 3, 6) = 2/3 which is the same.