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Chapter 4: Problem 19

Which of the following values cannot be probabilities of events and why? \(\begin{array}{llllllll}1 / 5 & .97 & -.55 & 1.56 & 5 / 3 & 0.0 & -2 / 7 & 1.0\end{array}\)

Short Answer

Expert verified
-0.55, 1.56, 5/3, and -2/7 do not represent valid probabilities as they do not fall within the range of 0 to 1 inclusive.

Step by step solution

01

Recognize the valid range

Start by knowing that the valid range of probability is between and including 0 and 1. That is, \(0 \leq P(E) \leq 1\), where P(E) denotes the probability of an event E.
02

Identify the values within the valid probability range

Find the values from the list which fall inside the valid range. The values are 1/5 (or 0.2), 0.97, 0.0, and 1.0.
03

Identify the values outside the valid probability range

Identify the values from the list which fall outside the valid range. These would be values that are less than 0 and greater than 1. The values are -0.55, 1.56, 5/3 (or 1.67), and -2/7 (or -0.2857).

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Key Concepts

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

Valid probability range
Probability is a fascinating concept! However, it's crucial to understand that probabilities must adhere to a valid range. This is the first principle you should always remember. The valid probability range is between 0 and 1, inclusive.
Simply put, this means any probability must be equal to or larger than 0 and equal to or smaller than 1. These numbers represent the degree of certainty of an event occurring.
  • If a probability is 0, it is certain that the event will not occur.
  • If a probability is 1, it is certain that the event will occur.
Understanding this range is fundamental, as it guides us when determining if numbers can indeed be probabilities.
Probability of an event
Probability tells us how likely an event is to happen. Think of it as the chances we attribute to an event occurring. When we talk about the probability of an event, we express it as a fraction, decimal, or percentage. All of these forms make it easy to comprehend the likelihood of even complex events.
For example, with a fraction such as \(\frac{1}{5}\), we express it as a decimal: 0.2. This tells us there is a 20% chance of the event happening 鈥� neat, isn't it?
It's important to convert different forms into decimals to make quick and easy comparisons with the confirmed range of probabilities.
Identifying valid probabilities
To identify whether a value could be a probability of an event, check if it falls within the valid range from 0 to 1. This is a vital step.
  • Values such as 0.0, 0.2 (\(\frac{1}{5}\)), 0.97, and 1.0 directly fall within this range.
  • Each of these demonstrates feasible scenarios where certainty or possibility exists for an event.
Always examine probabilities against the range criterion to confirm their validity.
Values outside probability range
Sometimes you will encounter values that fall outside the acceptable probability range.
It's essential to spot these errors immediately to avoid confusion. Values like -0.55, 1.56, or even \(\frac{5}{3}\) (or 1.67) are beyond the permissible threshold and cannot be probabilities.
  • Negative values are illogical in the context of probabilities, as they imply less than certain non-occurrence.
  • Values above 1 suggest more certainty than is possible in probability alone, breaching fundamental principles.
Recognizing these missteps will ensure that you have a strong grasp on valid probabilities.

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Most popular questions from this chapter

Five hundred employees were selected from a city's large private companies, and they were asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared $$\begin{array}{llc} \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array}$$ Suppose one employee is selected at random from these 500 employees. Find the following probabilities. a. The probability of the union of events "woman" and "yes" b. The probability of the union of events "no" and "man'

A company is to hire two new employees. They have prepared a final list of eight candidates, all of whom are equally qualified. Of these eight candidates, five are women. If the company decides to select two persons randomly from these eight candidates, what is the probability that both of them are women? Draw a tree diagram for this problem.

A restaurant menu has four kinds of soups, eight kinds of main courses, five kinds of desserts, and six kinds of drinks. If a customer randomly selects one item from each of these four categories, how many different outcomes are possible?

In a group of 10 persons, 4 have a type A personality and 6 have a type B personality. If two persons are selected at random from this group, what is the probability that the first of them has a type A personality and the second has a type B personality? Draw a tree diagram for this problem

The probability that a student graduating from Suburban State University has student loans to pay off after graduation is \(.60\). If two students are randomly selected from this university, what is the probability that neither of them has student loans to pay off after graduation?
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