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Descriptive Statistics and Basic Probability

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Descriptive Statistics and Basic Probability
Complementary Events

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Complementary Events

An event and its complement ("not" the event) always have probabilities that add up to exactly 1.

Everything That Isn't A

Term

$Complementary events$

Exact definition

Event A and "not A" (everything in the sample space that is not A) are complementary events: P(A) + P(not A) = 1, so P(not A) = 1 - P(A).

In plain English

"A happens" and "A doesn' t happen" always add up to a probability of 1 together.

Example

If P(rain) = 0.3, then P(no rain) = 1 - 0.3 = 0.7 .

Example 1 — finding a complementary probability

The probability it rains tomorrow is 0.3 . Find the probability it does NOT rain. 1 Use P(not A) = 1 - P(A) 2 1 - 0.3 Answer: 0.7

Example 2 — a second example

A bag has P(drawing red) = 0.4 . Find P(not red). 1 1 - 0.4 Answer: 0.6

Watch this mistake

Forgetting that complementary probabilities must sum to exactly 1

Stating P(A) = 0.3 and P(not A) = 0.6 for the same event, without checking the two values actually add to 1 .

Fix

Do this instead

P(A) and P(not A) must always add up to exactly 1 — if they don' t, something has gone wrong in the calculation.

Practice

The probability of passing an exam is 0.85 . Find the probability of failing.

Your answer

Take this with you

Before you go on

$P(A) + P(not A) = 1, always.$
$To find the complement of an event, subtract its probability from 1 .$

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