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Chapter 8: Problem 1

Does the number represent a probability? $$ \frac{11}{13} $$

Short Answer

Expert verified
Yes, \(\frac{11}{13}\) is a valid probability, as it falls within the range 0 to 1.

Step by step solution

01

Understand Probability Values

Probabilities are values that indicate the likelihood of an event occurring and must range from 0 to 1. A probability of 0 represents an impossible event, while a probability of 1 represents a certain event.
02

Convert Fraction to Decimal Form

To better assess if the fraction \(\frac{11}{13}\) represents a valid probability, we should convert it to a decimal. Perform the division: \(11 \div 13 \approx 0.846\).
03

Check the Range of the Decimal

Compare the decimal value 0.846 to the range 0 to 1. Since 0.846 is greater than 0 and less than 1, it falls within the valid range for probability values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Likelihood of an event
In the world of probability, the likelihood of an event is essentially a measure of how probable it is for a specific outcome to occur. This measure is expressed as a probability value鈥攁 number that reflects the chance of an event happening.
Probability values simplify the communication of uncertainty because they provide a precise way to represent likelihood.

Here are some key characteristics about event likelihood in probability:
  • A probability of 0 means the event is impossible and will not occur.
  • A probability of 1 means the event is certain to occur and is inevitable.
  • A probability between 0 and 1 indicates a gradation of likelihood, where the closer the number is to 1, the more likely the event is to happen.
This concept allows us to quantify uncertainty in practical terms. Whether you're rolling dice or predicting the weather, understanding the likelihood of an event is pivotal in decision-making.
Range of probability values
Probability has a specific range of values that one can use to determine if a given number could indeed be a valid probability. This range extends from 0 to 1.
Any number outside of this interval cannot represent a probability as it would imply either an impossible scenario (<0) or something beyond certainty (>1).

Let's look at this range concept more closely:
  • Values in the range \(0 \, \text{to} \, 1\) give a clear reflection of possible outcomes.
  • Every conceivable event will have a probability value positioned within this spectrum.
  • The endpoints 0 and 1 correspond to impossibility and certainty, anchoring the scale that probabilities use.
Recognizing the valid range ensures that calculations and assumptions made based on probability are grounded in statistical reality.
Fraction to decimal conversion
Understanding how to convert fractions to decimals is crucial in probability problems. This conversion helps us deal with numbers conveniently, especially when you need to compare values or fit them within a certain range.
For instance, determining if a fraction represents a valid probability can be simplified by converting it to its decimal form.

Here's how the conversion process works:
  • Take the fraction \(\frac{11}{13}\).
  • Divide 11 by 13 using long division or a calculator to get approximately 0.846.
  • Now, compare this decimal form against the known probability range of 0 to 1.
In our example, 0.846 is a clear fit within the range, confirming that \(\frac{11}{13}\) indeed represents a valid probability. Converting fractions to decimals helps streamline the analysis of probabilities and enhances our understanding of these concepts.

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Most popular questions from this chapter

Use the binomial theorem to expand each expression. $$ \left(p^{2}-3\right)^{4} $$

Suppose an employee's initial salary is 30,000 dollar. (a) If this person receives a 2000 dollar raise for each year of experience, determine a sequence that gives the salary at the beginning of the \(n\) th year. What type of sequence is this? (b) Suppose another employee has the same starting salary and receives a \(5 \%\) raise after each year. Find a sequence that computes the salary at the beginning of the \(n\) th year. What type of sequence is this? (c) Which salary is higher at the beginning of the 10 th year and the 20 th year? (d) Graph both sequences in the same viewing rectangle. Compare the two salaries.

Use Pascal's triangle to help expand the expression. $$ (2 a+3 b)^{3} $$

Use the binomial theorem to expand each expression. $$ (m+2)^{3} $$

Find the specified term. The fourth term of \((a+b)^{9}\)
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