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What Are Complementary Events in Probability?

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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 $\overset{}{A}$ . $A$ and $A$ with a bar on top (“A Bar”) are then complementary events. So you know that:

Rule

Complementary Events

P (A) + P (\overset{}{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 (\geq 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 (\geq 2) = 1 - P (1) = 1 - \frac{1}{6} = \frac{5}{6}

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

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What Are Disjoint Events in Probability?

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