Problem 57 - College Algebra

Question

Explain how to find the probability of an event not occurring. Give an example.

Step-by-Step Solution

Answer

The probability of an event not occurring, known as the complement, is given by subtracting the probability of the event from 1, expressed mathematically as \(1 - P(E)\) where \(P(E)\) is the probability of the event occurring. In the case of a coin toss, if the probability of getting a Head (H) is 0.5, the probability of not getting a Head, therefore getting a Tail (T), is \(1 - 0.5 = 0.5\).

Step 1: Understanding Probabilities

Probability is a measure of the likelihood or chance that a particular event will occur. Probability values are between 0 and 1. A probability of 1 means the event will definitely occur, while a probability of 0 means the event will not occur. If an event has a probability of \(P(E)\), it's complement, the event not occurring, is represented by \(1 - P(E)\) or \(P(E')\).

Step 2: Complementary Events

Two events are said to be complementary if one of the events occurs if and only if the other does not. The sum of the probabilities of an event and its complementary event is always equal to 1. This is due to the total probability law, which states that the sum of the probabilities of all possible outcomes of an event is 1. In this exercise, the complementary event is the event not occurring. Hence the probability of an event not happening is calculated as \(1 - P(E)\), with \(P(E)\) being the probability of the event occurring.

Step 3: Example

Let's take a real life example to illustrate this concept. If we toss a fair coin, the probability of Head (H) coming up is \(1/2\) or \(0.5\). This is because there are two possible outcomes, Head (H) and Tail (T), and we are looking at one specific outcome (H). So the probability of the event not happening, which in this case is getting a Tail (T), is calculated as \(1 - P(H)\). Thus, \(P(T) = 1 - P(H) = 1 - 0.5 = 0.5\). So the likelihood of getting a Tail (T) is also \(0.5\), which means that there's an equal chance of getting either a Head (H) or a Tail (T) when a fair coin is tossed.