Problem 35 - Precalculus with Limits

Question

In Exercises 35 - 38, you are given the probability that an event will happen. Find the probability that the event will not happen. \( P(E) = 0.87 \)

Step-by-Step Solution

Answer

The probability that the event E will not happen, or the probability of E', is 0.13.

Step 1: Understand the Concept of Complementary Events

The concept of complementary events in probability theory is about the occurrence or non-occurrence of an event. If 'E' is an event, then the 'not E' is called the complementary event of E (can be denoted as E'). The total probability of the occurrence and non-occurrence of an event is always 1.

Step 2: Deduce the Formula for Complement Probability

The probability of the occurrence of a complementary event E' is calculated by subtracting the probability of event E from 1, as follows: \( P(E') = 1 - P(E) \)

Step 3: Substituting in Formula

Substitute the given probability of the event into the complement probability formula. Thus, \( P(E') = 1 - 0.87 \)

Step 4: Calculate the Probability

Simple subtraction is now needed. When you subtract 0.87 from 1, the result is 0.13

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides a mathematical framework for predicting the outcomes of random phenomena and assessing the chances of different scenarios. At its most basic level, probability can be viewed as the ratio of the number of favorable outcomes to the total number of possible outcomes.

For instance, if you are flipping a fair coin, there are two possible outcomes: heads or tails. Each outcome has an equal chance of happening, so the probability of getting heads is 0.5, as is the probability of getting tails. Simplifying complex events and understanding the likelihood of various outcomes allows us to make informed decisions in uncertainty.

Complement Probability Formula

The complement probability formula expresses a fundamental relationship in probability theory: the sum of the probabilities of an event and its complement is always equal to one. The 'complement' of an event refers to all other possible outcomes that are not included in the original event. Essentially, it's the probability that the event does not occur.

To find the complement of an event, you simply subtract the probability of the event from 1, using the formula: \[ P(E') = 1 - P(E) \]
In the context of our exercise, if the event E occurs with a probability of 0.87, then the probability of 'not E' is calculated as follows: \[ P(E') = 1 - 0.87 = 0.13 \].

Subtracting Probabilities

Subtracting probabilities is a common technique used in calculating the likelihood that certain events will not happen. This technique is particularly useful when dealing with complementary events, where one event occurring necessarily means the other cannot.

For example, if you’re given the probability of it raining today is 0.30, the probability of it not raining is calculated by subtracting 0.30 from 1, giving you a probability of 0.70 for no rain. It is a straightforward calculation but immensely powerful for understanding the complete picture of possible outcomes in a scenario.