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Chapter 5: Problem 10

Which of the following numbers could not be probabilities, and why? a. \(38.4\) b. \(3.84 \%\) c. \(-3.84\) d. \(384 \%\) e. \(0.00384\)

Short Answer

Expert verified
Numbers a. \(38.4\), c. \(-3.84\) and d. \(384\%\) cannot be probabilities as they do not follow the rules of probabilities. Numbers b. \(3.84\%\) and e. \(0.00384\) are valid probabilities.

Step by step solution

01

Check Number A

A number indicates a probability if it falls between 0 and 1 (when not in percentage form). The number a. \(38.4\) is greater than 1, therefore it cannot be a probability.
02

Check Number B

The number b. \(3.84\%\) is a percentage and falls within the 0 to 100% range, which makes this a valid probability.
03

Check Number C

Probabilities must be non-negative. Therefore, the number c. \(-3.84\) is not a probability since it is a negative number.
04

Check Number D

As percentages, probabilities should range between 0% to 100%. The number d. \(384\%\) is greater than 100%, therefore it cannot be probability.
05

Check Number E

Lastly, the number e. \(0.00384\) lies between 0 and 1, making it a valid probability.

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Key Concepts

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

Probability Range
In probability theory, it is crucial to understand the valid range of numbers that can represent a probability. A valid probability must always lie between 0 and 1, inclusive.
This can be visualized as:
  • 0, representing an impossible event, like rolling a 7 on a standard six-sided die.
  • 1, representing a certain event, such as getting either a head or a tail when flipping a coin.
Any number outside this range cannot be a probability because it does not conform to the mathematical definition of probability. This means numbers less than 0 or greater than 1 are invalid as probabilities.
This is often a key point of confusion for students, especially when probability is presented in various formats such as decimals, fractions, or percentages.
Percentage to Probability Conversion
Probabilities are often represented as percentages, which require conversion for easier interpretation.
When confronted with a percentage as a probability, it must first be converted to a decimal format to see if it falls within the valid probability range of 0 to 1.
Here's how you can convert percentages to probabilities:
  • Move the decimal point two places to the left.
  • For example, to convert 3.84% to a probability, move the decimal to get 0.0384, which is a valid probability as it lies between 0 and 1.
Remember that any percentage greater than 100% is not a valid probability, because once converted, it results in a number greater than 1. Thus, these cannot be considered probabilities.
Negative Probabilities
Negative numbers present in a probability problem often confuse learners, but it's quite simple when you remember a fundamental rule. Probabilities are always non-negative.
This means that under no circumstances can you have a valid probability that is negative.
Negative probabilities suggest events that are even less possible than impossible, which does not make sense within the realm of probability.
  • For instance, a probability of ( -3.84 ) is clearly invalid, since probabilities cannot be less than 0.
Remember this simple yet important rule: if a number is negative, it cannot function as a probability.
Understanding Valid Probabilities
Recognizing valid probabilities is fundamental when interpreting statistical data and analyzing outcomes.
Practically speaking, if a number lies within 0 to 1, it haves the potential to be a probability. For example, 0.00384, though a small probability, is still valid since it lies within this range.
Here are some key pointers:
  • Always check if the number is within 0 to 1 when in decimal form, or 0% to 100% when in percentage form.
  • Understand that sometimes 1 is included in the range, allowing for a 100% certainty possibility.
Being comfortable with these concepts helps you correctly identify which numbers can truly act as probabilities, thereby drawing more accurate conclusions from data that involves uncertainty.

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Most popular questions from this chapter

State Assembly A state assembly is supposed to represent the population. We wish to perform a simulation to determine an empirical probability that an assembly of 50 representatives has 25 or fewer males. Assume that about \(50 \%\) of the population is male, so the probability that a person who has been elected to the state assembly is a male is \(50 \%\). Using a random number table, we decide that each digit will represent an assembly member. The digits \(0-25\), we decide, will represent a male member, and \(26-50\) will represent a female. Why is this a bad choice for this simulation?

Thumbtacks When a certain type of thumbtack is tossed, the probability that it lands tip up is \(60 \%\), and the probability that it lands tip down is \(40 \%\). All possible outcomes when two thumbtacks are tossed are listed. U means the tip is Up, and D means the tip is Down. \(\begin{array}{llll}\text { UU } & \text { UD } & \text { DU } & \text { DD }\end{array}\) a. What is the probability of getting exactly one Down? b. What is the probability of getting two Downs? c. What is the probability of getting at least one Down (one or more Downs)? d. What is the probability of getting at most one Down (one or fewer Downs)?

Internet Access A 2013 Pew poll said that \(93 \%\) of young adults in the United States have Internet access. Assume that this is still correct. \(\mathrm{a} .\) If two people are randomly selected, what is the probability that they both have Internet access? b. If the two people chosen were a married couple living in the same residence, explain why they would not be considered independent with regard to Intemet access.

GSS: Political Party The General Social Survey (GSS) is a survey done nearly every year at the University of Chicago. One survey, summarized in the table, asked each respondent to report her or his political party affiliation and whether she or he was liberal, moderate, or conservative. (Dem stands for Democrat, and Rep stands for Republican.) $$ \begin{array}{lcccc} & \text { Dem } & \text { Rep } & \text { Other } & \text { Total } \\ \hline \text { Liberal } & 306 & 26 & 198 & 530 \\ \hline \text { Moderate } & 279 & 134 & 322 & 735 \\ \hline \text { Conservative } & 104 & 309 & 180 & 593 \\ \hline \text { Total } & 689 & 469 & 700 & 1858 \end{array} $$ a. If one person is chosen randomly from the group, what is the probability that the person is liberal? b. If one person is chosen randomly from the group, what is the probability that the person is a Democrat?

Seat Belt Use In 2009 , the National Highway Traffic Safety Administration said that \(84 \%\) of drivers buckled their seat belts. Assume that this percentage is still accurate. If four drivers are randomly selected, what is the probability that they are all wearing their seat belts?
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