Wolfram Alpha:

Probability Rules
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Mutually Exclusive Events
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A and B cannot happen at the same time.  Therefore:

P(A and B) ≡ P(A ∩ B) = 0

Example:

A simple coin toss.

Independent Events
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A and B are not related to each other.  Therefore:

P(A ∩ B) = P(A).P(B)

Clearly A and B cannot be both mutually exclusive and independent.

Example:

Probability of rolling a 6 and tossing a head is (1/6)(1/2) = 1/2

Dependent Events
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When A and B, are dependent, the probability of both occurring is:

P(A ∩ B)  =  P(A)·P(B|A)

Example:

A card is chosen at random from a deck of cards. Without replacing
it, a second card is chosen. What is the probability that the first
card chosen is a queen and the second card chosen is a jack?

P(Q) = 4/52

P(J|Q) = 4/51

P(Q ∩ J) = (4/52)(4/51) = 4/663

Dependent events are discussed in more detail in the note on
conditional probability.

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If A and B ARE mutually exclusive:

P(A or B) ≡ P(A ∪ B) = P(A) + P(B)

Example:

A single die is rolled. What is the probability of rolling a
2 or a 5?

P(2 ∪ 5) = P(2) + P(5)

= 1/6 + 1/6

= 1/3

If A and B are NOT mutually exclusive:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

= P(A) + P(B) - P(A).P(B)

= P(A) + P(B)(1 - P(A))

Example:

What is the probability pulling a club or a king from a deck
of cards?

P(C ∪ K) = P(C) + P(K) - P(C ∩ K)

= 4/52 + 13/52 - 1/52 or 4/52 + (13/52)(1 - 4/52 )

= 4/13