Also....if two events A and B are independent, and you know that P(A) = 0.3, what is the value of P(A|B)?
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By definition, mutually exclusive events cannot occur at the same time. The prototypical example is rolling a single die (plural: dice). All the possible outcomes, {1, 2, ... , 6}, are mutually exclusive events, so P(A OR B) = P(A) + P(B). For example, P(2 or 6) = P(2) + P(6) = 1/6 + 1/6 = 1/3.
An example of non-mutually exclusive events is when rolling 2 dice - here you can have A and B simultaneously, for example rolling a 2 and a 6 together. In this situation you must calculate:
P(A OR B) = P(A) + P(B) - P(A AND B).
For example:
P(2 or 6) = P(2) + P(6) - P(2 and 6) = 11/36 + 11/36 - 2/36 = 20/36
Observe that when A and B are mutually exclusive, P(A AND B) = 0, so the rule
P(A OR B) = P(A) + P(B) - P(A AND B) works for all cases, mutually exclusive or not.
When events are not mutually exclusive, we ask whether they are independent. By definition, independent events have the property that P(A AND B) = P(A)P(B).
For example, the outcomes of rolling two dice do not affect each other. Therefore, P(2 AND 6) = P(2)P(6) = 1/6 x 1/6 = 1/36.
Note: P(2 AND 6) and P(6 AND 2) are mutually exclusive events, if you want to include both of them, add their probabilities ( = 2/36).
The formula for conditional probability is:
P(A|B) = P(A AND B)/P(B)
If A and B are independent, then P(A AND B) = P(A)P(B) so,
P(A|B) = P(A)P(B) /P(B) = P(A), so the answer to your second question is P(A|B) = P(A) = 0.3.
An example of non-mutually exclusive events is when rolling 2 dice - here you can have A and B simultaneously, for example rolling a 2 and a 6 together. In this situation you must calculate:
P(A OR B) = P(A) + P(B) - P(A AND B).
For example:
P(2 or 6) = P(2) + P(6) - P(2 and 6) = 11/36 + 11/36 - 2/36 = 20/36
Observe that when A and B are mutually exclusive, P(A AND B) = 0, so the rule
P(A OR B) = P(A) + P(B) - P(A AND B) works for all cases, mutually exclusive or not.
When events are not mutually exclusive, we ask whether they are independent. By definition, independent events have the property that P(A AND B) = P(A)P(B).
For example, the outcomes of rolling two dice do not affect each other. Therefore, P(2 AND 6) = P(2)P(6) = 1/6 x 1/6 = 1/36.
Note: P(2 AND 6) and P(6 AND 2) are mutually exclusive events, if you want to include both of them, add their probabilities ( = 2/36).
The formula for conditional probability is:
P(A|B) = P(A AND B)/P(B)
If A and B are independent, then P(A AND B) = P(A)P(B) so,
P(A|B) = P(A)P(B) /P(B) = P(A), so the answer to your second question is P(A|B) = P(A) = 0.3.
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If they are mutually exclusive, either one or the other happens at one time.
If they are independent, both can happen at the same time
If they are independent, both can happen at the same time
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Only if you apply Murphy's Law.
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Not if they are happening at the same time.....