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Probability Notes: Negation Rule, Addition Rule, Multiplication Rule

Negation Rule

For event A,

P(not A) = 1 - P(A)

Additional Rule

For events A and B:

P(A or B) = P(A) + P(B) - P(A and B)

If the events are multually exclusive, the rule simplifies to

P(A or B) = P(A) + P(B)

Multiplication Rule for Independent Events

For two independent events A and B,

P(A and B) = P(A) × P(B)

Three Dice Example

What is the probability, if you roll three dice, that at least one dice will be a 6?

The simplest approach is to use the negation rule with the multiplication rule. It is easy to calculate the probability for NOT getting a 6 in a roll, which is 5/6. And therefore The probability for NOT getting a 6 in three rolls is (5/6)³. And the 1 - (5/6)³ ≅ 0.421 or 42.1%.

A slightly more complicated approach is to use the additional rule and the multiplication rule iteratively. First we calculate the probability of rolling a six in either the first or the second dice:

P(A or B) = P(A) + P(B) - P(A and B) = 1/6 + 1/6 - (1/6)×(1/6) = 11/36

Then, we calculate again including the third dice:

P([A or B] or C) = P(A or B) + P(C) - P([A or B] and C)

= 11/36 + 1/6 - (11/36)×(1/6) ≅ 0.421 or 42.1%