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, or the complement, can be found by subtracting the probability of the event from 1. For instance, in a standard deck of cards, the probability of not drawing a Heart is 0.75.

Step 1: Understand the Complement Rule

Firstly, understand the complement rule in probability. The complement of an event is the event not happening. So, the sum of the probabilities of an event and its complement is always 1. Thus, if 'A' is an event, then the probability of 'A' not occurring, denoted as \(P(A')\) or \(P(A^c)\), is 1 minus the probability of 'A' occurring i.e., \(P(A' )=1-P(A)\).

Step 2: Parameter Identifcation

Let's illustrate this with an example. Suppose we have a standard deck of 52 playing cards, and we draw one randomly. Let's say event 'A' is drawing a card that is a Heart. There are 13 Hearts in the deck, so the probability of drawing a Heart, \(P(A)\), is \(\frac{13}{52}=0.25\).

Step 3: Applying the Complement Rule

Now, to find the probability of event 'A' not occurring (i.e., not drawing a Heart), we can use the complement rule, that is \(P(A')=1-P(A)\). So, subtract the probability of drawing a Heart from 1, resulting in \(P(A')=1-0.25=0.75\). So, the probability of not drawing a Heart from a standard deck of cards is 0.75.

Step 4: Final Statement

So, we've clearly outlined how to calculate the probability of an event not happening (the complement) using an example with playing cards. This principle can be generalized to any probabilistic scenario.