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Chapter 4: Problem 15

Briefly explain the two properties of probability.

Short Answer

Expert verified
The two properties of probability are Non-negativity, which states that the probability of any event is always nonnegative (i.e., \(P(A) \geq 0\) for every event A), and Normalization, which states that the probability of a sure event is 1 (i.e., \(P(S) = 1\), where S is the set of all possible outcomes).

Step by step solution

01

Explain Non-negativity

In Probability, the first fundamental property is Non-negativity. Non-negativity states that the probability of any event is always nonnegative, that is, it should always be 0 or greater. It is depicted as: \[P(A) \geq 0 \] for every event A.
02

Explain Normalization

The second property to be discussed is normalization. The normalization property states that the probability of a sure event (the set of all possible outcomes) is 1. In other words, something is certain to happen. This is expressed as, \[P(S) = 1 \] where S represents the sample space, or set of all possible outcomes.

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Most popular questions from this chapter

In a sample survey, 1800 senior citizens were asked whether or not they have ever been victimized by a dishonest telemarketer. The following table gives the responses by age group (in years). $$\begin{array}{l|lccc} & & & \begin{array}{c} \text { Have Been } \\ \text { Victimized } \end{array} & \begin{array}{c} \text { Have Never } \\ \text { Been Victimized } \end{array} \\ \hline & 60-69 & \text { (A) } & 106 & 698 \\ \text { Age } & 70-79 & \text { (B) } & 145 & 447 \\ & 80 \text { or over } &\text { (C) } & 61 & 343 \\ \hline \end{array}$$ a. Suppose one person is randomly selected from these senior citizens. Find the following probabilities. i. \(P(\) have been victimized and \(\mathrm{C}\) ) ii. \(P(\) have never been victimized and \(\mathrm{A}\) ) b. Find \(P(B\) and \(C\) ). Is this probability zero? Explain why or why not.

A restaurant chain is planning to purchase 100 ovens from a manufacturer, provided that these ovens pass a detailed inspection. Because of high inspection costs, 5 ovens are selected at random for inspection. These 100 ovens will be purchased if at most 1 of the 5 selected ovens fails inspection. Suppose that there are 8 defective ovens in this batch of 100 ovens. Find the probability that the batch of ovens is purchased. (Note: In Chapter 5 you will learn another method to solve this problem.)

Five percent of all items sold by a mail-order company are returned by customers for a refund. Find the probability that, of two items sold during a given hour by this company, a. both will be returned for a refund b. neither will be returned for a refund Draw a tree diagram for this problem.

The probability that a randomly selected college student attended at least one major league baseball game last year is .12. What is the complementary event? What is the probability of this complementary event?

Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two- way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same as. or worse off than their parents. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than } \\ \text { High School } \end{array} & \begin{array}{c} \text { High } \\ \text { School } \end{array} & \begin{array}{c} \text { More Than } \\ \text { High School } \end{array} \\ \hline \text { Better off } & 140 & 450 & 420 \\ \text { Same as } & 60 & 250 & 110 \\ \text { Worse off } & 200 & 300 & 70 \end{array}$$ Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. a. \(P(\) better off or high school \()\) b. \(P(\) more than high school or worse off \()\) c. \(P\) (better off or worse off)
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