Problem 58 - College Algebra

Question

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

Step-by-Step Solution

Answer

To find the probability of an event not occurring, first find the probability of the event occurring and then subtract this from 1. For instance, if the probability of getting a '4' in a dice throw is \( \frac{1}{6} \), the probability of not getting a '4' is \( 1 - \frac{1}{6} = \frac{5}{6} \).

Step 1: Understand Probability

Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1. A Probability of 1 indicates certainty the event will occur, and 0 indicates the event will not occur. Probability can be calculated by dividing the number of favorable outcomes by the total number of outcomes.

Step 2: Define the Event

An event is any outcome or combination of outcomes. For this example, consider a simple event of tossing a fair six-sided dice. The event can be getting a '4'. There are 6 possible outcomes (1, 2, 3, 4, 5, 6) and only one favorable outcome which is '4'.

Step 3: Calculate Probability of Event Occurring

To calculate the probability of '4' occurring, divide the number of favorable outcomes (1, since '4' is just one number) by the total number of outcomes (6, since there are six faces on the dice). The probability is \( \frac{1}{6} \).

Step 4: Calculate Probability of Event Not Occurring

The understanding that the sum of probabilities of all possible exhaustive outcomes is 1, which is the Principle of Complementarity. To get the probability of the event not occurring (not getting a '4') is calculated as 1 minus the probability of the event (getting a 4). Therefore, the probability of not getting a '4' is \( 1 - \frac{1}{6} = \frac{5}{6} \).