7.3 Theorem of Complementary Events

The Complementary Theorem in probability is a fundamental concept used to find the probability of an event not happening. It states that the sum of the probabilities of an event and its complement is always equal to 1.

What Is a Complementary Event?

Two events are considered complementary if:

1. One of them must happen: Either the event occurs, or it does not. There are no other possibilities.
2. They cannot both happen at the same time: An event and its complement are mutually exclusive.

In simple terms, if event 'A' is an event, its complement, denoted as 'A' or 'Ac', represents all the outcomes that are not in event A. The total probability of both outcomes (A happening or A not happening) must sum up to 1.

The relationship can be expressed as:

$P(A) + P(A') = 1$

Where:

• $P(A)$ is the probability that event A occurs.
• $P(A')$ is the probability that event A does not occur (the complement of A).

This also leads to the primary formula for calculating the probability of a complementary event:

$P(A') = 1 - P(A)$

Complementary Events Are Mutually Exclusive and Collectively Exhaustive

• Mutually Exclusive: Events A and A' cannot occur simultaneously. For example, you cannot get both heads and tails on a single coin flip.
• Collectively Exhaustive: The union of event A and its complement A' covers all possible outcomes in the sample space. This means that either A will happen, or its complement A' will happen.

Formula for Complementary Probability

The core formula used to calculate the probability of a complementary event is:

$P(A') = 1 - P(A)$

This formula is particularly useful when the probability of the event occurring, $P(A)$, is easier to calculate than the probability of the event not occurring, $P(A')$.

Examples

Example 1: Coin Flip

Problem: Suppose we flip a fair coin once. Let event A be “getting Heads”. What is the probability of not getting Heads?

Solution: For a fair coin, there are two equally likely outcomes: Heads (H) and Tails (T). The sample space is {H, T}.

• Event A: Getting Heads.
• $P(A) = \text{Probability of getting Heads} = \frac{1}{2}$

The complement of event A, denoted as A', is “not getting Heads”, which means “getting Tails”.

Using the complement rule: $P(A') = 1 - P(A)$ $P(A') = 1 - \frac{1}{2}$ $P(A') = \frac{1}{2}$

Therefore, the probability of not getting Heads (i.e., getting Tails) is $1/2$.

Example 2: Drawing Balls from a Bag

Problem: A bag contains 5 red balls and 3 blue balls. One ball is drawn at random. Let event A be “drawing a red ball.” Find the probability of not drawing a red ball.

Solution: Total number of balls in the bag = 5 (red) + 3 (blue) = 8. The sample space consists of these 8 balls.

• Event A: Drawing a red ball.
• Number of red balls = 5.
• $P(A) = \text{Probability of drawing a red ball} = \frac{\text{Number of red balls}}{\text{Total number of balls}} = \frac{5}{8}$

The complement of event A, A', is “not drawing a red ball.” This means drawing a ball that is not red, which in this case, is drawing a blue ball.

Using the complement rule: $P(A') = 1 - P(A)$ $P(A') = 1 - \frac{5}{8}$ $P(A') = \frac{8}{8} - \frac{5}{8}$ $P(A') = \frac{3}{8}$

Therefore, the probability of not drawing a red ball (i.e., drawing a blue ball) is $3/8$.

Summary

Frequently Asked Questions (FAQs)

• What is the Complementary Theorem of Probability? The Complementary Theorem states that the probability of an event happening plus the probability of that event not happening is always equal to 1. It's expressed as $P(A) + P(A') = 1$.

• Define complementary events with an example. Complementary events are two events that are mutually exclusive (cannot happen at the same time) and collectively exhaustive (one of them must happen). For example, when rolling a standard six-sided die, the event "rolling a 4" and the event "not rolling a 4" are complementary.

• Why are complementary events mutually exclusive and collectively exhaustive? They are mutually exclusive because an event and its complement cannot both occur for the same outcome. They are collectively exhaustive because, by definition, a complementary event covers all possible outcomes not included in the original event, ensuring that one of the two must occur.

• State the formula for finding the probability of a complementary event. The formula is $P(A') = 1 - P(A)$, where $P(A')$ is the probability of the complement of event A, and $P(A)$ is the probability of event A.

• Explain the difference between an event and its complement. An event is a specific outcome or set of outcomes from a random experiment. Its complement includes all outcomes from the same experiment that are not part of the original event.

• If $P(A) = 0.7$, what is $P(A')$? Using the formula $P(A') = 1 - P(A)$, we get $P(A') = 1 - 0.7 = 0.3$.

• A coin is flipped once. What is the probability of not getting heads? The probability of getting heads ($P(\text{Heads})$) is $0.5$. The probability of not getting heads ($P(\text{Not Heads})$) is $1 - 0.5 = 0.5$.

• A die is rolled once. What is the probability of not getting a 6? The probability of getting a 6 ($P(6)$) is $1/6$. The probability of not getting a 6 ($P(\text{Not } 6)$) is $1 - 1/6 = 5/6$.

• In a bag with 4 green and 6 yellow balls, find the probability of not drawing a green ball. Total balls = 10. Probability of drawing a green ball ($P(\text{Green})$) = $4/10 = 2/5$. The probability of not drawing a green ball ($P(\text{Not Green})$) is $1 - P(\text{Green}) = 1 - 2/5 = 3/5$.

• If the probability of rain tomorrow is 0.3, what is the probability it will not rain? Let A be the event "it rains tomorrow". $P(A) = 0.3$. The probability it will not rain ($P(A')$) is $1 - P(A) = 1 - 0.3 = 0.7$.