Problem 8 - College Algebra with Modeling and Visualization

Question

Does the number represent a probability? $$ \frac{9}{8} $$

Step-by-Step Solution

Answer

No, \( \frac{9}{8} \) does not represent a probability as it is greater than 1.

Step 1: Understand Probability

Probability is a measure of how likely it is for an event to occur. It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. Any number outside this interval cannot represent probability.

Step 2: Analyze the Fraction

Given the fraction \( \frac{9}{8} \), let's determine if it lies within the range of a valid probability, which is between 0 and 1.

Step 3: Convert the Fraction to Decimal

Divide the numerator by the denominator: \( 9 \div 8 = 1.125 \).

Step 4: Evaluate the Range

Compare the decimal \(1.125\) with the probability range. Since \(1.125\) is greater than 1, it doesn't fall within the range of 0 to 1.

Step 5: Conclusion

Since the value \( \frac{9}{8} \) equals \(1.125\) which is outside the range of 0 to 1, it cannot represent a probability.

Key Concepts

Probability range

Probability is a fundamental concept in mathematics and statistics, helping us predict how likely an event is to happen. The range of probability is from 0 to 1. This means that for any event:

• A probability of 0 means the event is impossible. It cannot happen under any circumstances.
• A probability of 1 means the event is certain. It will definitely happen.
• A probability between 0 and 1 represents the event's likelihood of occurring, where closer to 0 is less likely and closer to 1 is more likely.

If a number lies outside this range, it cannot be a probability. Such numbers imply probabilities that are not possible given the boundaries established by the rules of probability.

Probability measure

Probability is measured to understand the chance of a specific event happening. The concept of probability measure applies mathematical principles to quantify uncertainty. For instance:

• If you flip a fair coin, the probability measure of it landing on heads is 0.5, denoting a 50% chance.
• In rolling a six-sided die, the probability measure of landing on any one face is \(\frac{1}{6}\), as each face has an equal chance.

When using a probability measure, we're assigning a real number between 0 and 1 to the likelihood of occurrence. This helps in a consistent assessment of events, guiding decisions and predictions based on those probabilities.

Fraction to decimal conversion

Converting fractions to decimals is a simple yet crucial step in evaluating probabilities. Sometimes, probabilities are given as fractions and need to be compared to the standard range or used in calculations. To convert a fraction like \(\frac{9}{8}\) into a decimal:

• Divide the numerator (9) by the denominator (8).
• Carry out the division \(9 \div 8 \) yielding 1.125.

This conversion is helpful because decimals are often easier to compare, especially when determining if a probability is within the permissible range of 0 to 1. In this context, if we convert \(\frac{9}{8}\) to 1.125, we immediately see it falls outside the range, thus it isn't a valid probability.