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What Are Complementary Events in Probability?

Complementary events A

A compound event A contains one or more of the outcomes in the sample space. The outcomes that are left are read as “not A”, and can be written as A¯. A and A with a dash on top are then complementary events. Then you know that:

Rule

Complementary Events

P (A) + P (A¯) = 1

Note! An important application is that

P (at least one) = 1 P (none)

Example 1

You roll a die once. What’s the probability of getting two dots or more on the die?

Here you’re looking to find the probability of getting 2, 3, 4, 5 or 6 dots on the die. The easiest way to calculate this is to use the application above:

P ( 2) = 1 P (< 2)

The event fewer than two dots is the same as getting one dot on the die, and the number of possibilities is then 1. That means you can write

P ( 2) = 1 P (1) = 1 1 6 = 5 6.

The probability of getting at least two dots when you roll a die is then 5 6 = 0.833.

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What Are Disjoint Events in Probability?