Problem 18 - Algebra for College Students

Question

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 is \( \frac{19}{20} \).

Step 1: Understand the Problem

We are given the probability that a certain horse will win the Kentucky Derby, which is \( \frac{1}{20} \). We need to find the probability that this horse will lose the race.

Step 2: Determine Total Probability

Probability is always out of 1, which means that the total probability of all possible outcomes (winning or losing) is 1.

Step 3: Use the Complement Rule

The probability of an event not happening is called the complement. In this case, the complement of the horse winning is the horse losing. The probability of the horse losing is \( 1 - \text{(probability of winning)} \).

Step 4: Calculate the Probability of Losing

Subtract the probability of the horse winning from 1: \( 1 - \frac{1}{20} \).

Step 5: Simplify the Calculation

Simplify the fraction: \( 1 - \frac{1}{20} = \frac{20}{20} - \frac{1}{20} = \frac{19}{20} \).

Step 6: Answer the Question

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

Key Concepts

Complement Rule

In probability, understanding the complement rule is essential. The complement of an event is basically everything that can happen outside of the specified event. If you know the probability of an event happening, the probability of it not happening is its complement.
To get the complement probability, subtract the probability of the event from 1, because probabilities must always add up to a whole, or 1 in decimal form. In our example, the probability of the horse winning the Derby is \( \frac{1}{20} \). Thus, the probability of the horse losing, the complement, is calculated as \( 1 - \frac{1}{20} \). This approach gives us the missing part of the probability spectrum.

Event Outcome

An event outcome in probability is any possible result that can occur in a given situation. In every probabilistic experiment, such as a horse race, there are one or more possible outcomes that are uncertain until the event actually happens. In our example, the outcomes are simplified to two main events: the horse wins or the horse loses.

• The probability that the horse wins is \( \frac{1}{20} \).
• The probability that the horse loses, as calculated, is \( \frac{19}{20} \).

These probabilities add up to 1, confirming that we've included all possible outcomes of the event.

Fraction Simplification

Fraction simplification is a core mathematical skill, especially in probability. It helps you to express results in the simplest form, ensuring easy understanding.
When subtracting fractions, the key is to have a common denominator, which stays the same throughout. In our exercise, we start with 1, which can be written as \( \frac{20}{20} \) to align it with the fraction \( \frac{1}{20} \).

• Original operation: \( 1 - \frac{1}{20} \)
• Express 1 as a fraction: \( \frac{20}{20} \)
• Subtract: \( \frac{20}{20} - \frac{1}{20} \)
• Simplified result: \( \frac{19}{20} \)

This approach helps clarify the final probability and makes solving similar problems much more manageable.