Problem 18 - Algebra for College Students

Question

Solve each problem.The probability that a certain horse will win the Kentucky Derby is \(\frac{1}{20}\). What is the probability that it will lose the race? \(\frac{19}{20}\)

Step-by-Step Solution

Answer

The probability that the horse will lose the race is \( \frac{19}{20} \).

Step 1: Understand the Problem

We have the probability of the horse winning the Kentucky Derby given as \( \frac{1}{20} \) and we are tasked with finding the probability of the horse losing the race.

Step 2: Recall the Complementary Rule of Probability

The sum of the probabilities of all possible outcomes must equal 1. Therefore, if we know the probability of one outcome (the horse winning), the probability of the other outcome (the horse losing) can be found by subtracting the given probability from 1.

Step 3: Apply the Complementary Rule

The probability that the horse loses the race is equal to 1 minus the probability that the horse wins the race. Therefore, we compute:\[P(\text{lose}) = 1 - P(\text{win}) = 1 - \frac{1}{20}\]

Step 4: Simplify the Calculation

Subtract \( \frac{1}{20} \) from 1:\[P(\text{lose}) = 1 - \frac{1}{20} = \frac{20}{20} - \frac{1}{20} = \frac{19}{20}\]Thus, the probability that the horse will lose the race is \( \frac{19}{20} \).

Key Concepts

Complementary Rule in Probability Theory

In probability theory, the complementary rule is a fundamental concept that helps simplify the computation of probabilities. When dealing with scenarios that have two possible outcomes, such as winning or losing a race, the complementary rule can be particularly useful. The rule states that the probabilities of all possible outcomes must add up to 1. This means if you know the probability of one outcome, you can easily find the probability of the other outcome by subtracting the known probability from 1. For instance, if you know the probability that a horse will win a race is \( \frac{1}{20} \), the complementary rule tells us that the probability of the horse losing is \( 1 - \frac{1}{20} \), which equals \( \frac{19}{20} \). This approach simplifies the process of solving many probability problems.

Understanding Probability Calculation

Probability calculation is a process used to determine the likelihood of a particular outcome. To calculate the probability, you divide the number of favorable outcomes by the total number of possible outcomes. The probability is usually expressed as a fraction, a decimal, or a percentage. For example, if a horse has a \( \frac{1}{20} \) chance of winning a race, it means there is 1 favorable outcome and 20 possible outcomes. This method helps in quantifying how likely an event is to occur, providing a numerical representation that ranges from 0 (impossible event) to 1 (certain event). The process of probability calculation is not only applicable to simple events but also extends to complex scenarios and is a cornerstone in the study of probability theory.

Probability of Outcomes

The concept of probability of outcomes is about understanding the different possible results of an event and how likely each is to occur. Every event has a set of possible outcomes, and probability helps us assign a likelihood to each of these. In the context of our horse race example, there are two primary outcomes: the horse wins, or the horse loses. By calculating the probability of each, we can make better predictions and understand the chances involved in an event. For instance, with a winning probability of \( \frac{1}{20} \), the complementary probability of not winning (losing) is \( \frac{19}{20} \). This reflects the complete set of possibilities, with both probabilities adding up to 1, showcasing a fundamental concept in probability theory: the sum of probabilities of all potential outcomes must equal 1.