Problem 12 - A Graphical Approach to Precalculus with Limits

Question

If the probability of an event is 0.857 what is the probability that the event will not occur?

Step-by-Step Solution

Answer

The probability that the event will not occur is 0.143.

Step 1: Understand Probability Complement

The probability of an event not occurring is called the complement of the probability of the event occurring. The sum of the probability of an event and its complement is always 1.

Step 2: Formula for Complement Probability

The formula to find the complement of an event is: \( P( ext{not A}) = 1 - P(A) \), where \( P(A) \) is the probability of the event occurring.

Step 3: Calculate the Complement Probability

Using the formula, substitute the given probability into the equation: \( P( ext{not A}) = 1 - 0.857 \).

Step 4: Perform the Calculation

Calculate \( 1 - 0.857 = 0.143 \). This is the probability that the event will not occur.

Key Concepts

Complementary Events

When we talk about complementary events, we refer to the scenario where one event occurring means that the other does not and vice-versa. Let's say we have an event A, its complement, denoted usually as \( A' \) or \( \text{not A} \), represents the possibility that event A does not happen.
Complementary events are quite intuitive: if you're rolling a die, the event "rolling an even number" automatically has the complementary event "rolling an odd number."
Certainly, the pair of an event and its complement will always cover all possible outcomes of a particular situation or experiment.

• If event A is "it rains today," then event \( A' \) is "it does not rain today."
• If event A is "a coin toss lands on heads," then event \( A' \) is "a coin toss lands on tails."

It is this concept that assures us when combined, the probabilities of an event and its complement must add up to 1.

Probability Formula

The formula for finding the probability of the complement of an event is both simple and powerful. To compute it, we use:
\[ P(\text{not A}) = 1 - P(A) \]
Here, \( P(A) \) represents the probability that event A does happen.
This formula is an expression of the total probability rule, grounded in the essential property of probabilities: the sum of the probability of all possible outcomes of an experiment equals 1.
When we use the formula, we harness this inherent symmetry of probabilities:

• Calculate the chance of the opposite situation happening by subtracting the known probability from 1.
• Use this to quickly assess the likelihood of not getting a desired outcome, which often aids in decision-making and planning.

By understanding and applying this formula, you can skillfully tackle a wide variety of probability problems.

Calculating Probability Complements

Now, let's apply what we've learned by calculating the complement probability, which is especially useful in real-world situations.
Imagine you know the probability of an event happening is 0.857, but what you really need is how likely it is for this event *not* to happen.
Here's how you can determine this quickly and effectively:

• Use the formula \( P(\text{not A}) = 1 - P(A) \).
• In this example, substitute \( P(A) = 0.857 \) into the formula.

Through simple subtraction, you find the complement:
\[ P(\text{not A}) = 1 - 0.857 = 0.143 \]
So, the probability that the event will not occur is 0.143.
This practice is common in everyday situations, such as risk assessment or predicting alternative outcomes. Understanding complements can provide comprehensive insights into scenarios by offering a fuller picture of possible events and their opposites.