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Chapter 4: Problem 20

Which of the following values cannot be probabilities of events and why? \(\begin{array}{llllllll}.46 & 2 / 3 & -.09 & 1.42 & .96 & 9 / 4 & -1 / 4 & .02\end{array}\)

Short Answer

Expert verified
The values -0.09, 1.42, 9/4 and -1/4 cannot be probabilities of events because they fall outside the range [0, 1].

Step by step solution

01

Analyzing the first value

The first value is 0.46. This number is between 0 and 1, thus it is a valid probability.
02

Analyzing the second value

The second value is 2/3 or approximately 0.67. This is also a number between 0 and 1, thus it is also a valid probability.
03

Analyzing the third value

The third value is -0.09. This number falls below 0 which is not in the valid range for probability values. Thus, -0.09 cannot be a probability.
04

Analyzing the fourth value

The fourth value is 1.42. This number is greater than 1, which is outside of the valid range for probability values. Hence, 1.42 cannot be a probability.
05

Analyzing the fifth value

The fifth value is 0.96. This number is between 0 and 1, making it a valid probability value.
06

Analyzing the sixth value

The sixth value is 9/4 or 2.25. This number is greater than 1, so it is outside of the valid range for probability values. Therefore, 9/4 cannot be a probability.
07

Analyzing the seventh value

The seventh value is -1/4 or -0.25. This number is less than 0, thus it is not a valid probability value. Hence, -1/4 cannot be a probability.
08

Analyzing the eighth value

The eighth value is 0.02. This number is between 0 and 1, hence it is a valid probability value.

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Key Concepts

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

Valid Probability
In probability theory, every event's likelihood is expressed as a probability value, which provides a numerical measure. For a probability to be considered valid, it must lie within a specific range. The value indicates how likely an event is to occur, where a valid probability can only be between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 suggests the event is certain to happen. Above 1, or below 0 probabilities are not valid since they do not align with probability theory principles.
Thus, understanding the concept of valid probability is vital for analyzing any statistical problem that involves measuring uncertainty.
Range of Probabilities
Understanding the range of probabilities is a core element in probability analysis.
The acceptable range is a simple linear scale from 0 to 1:
  • 0: Represents impossibility. The event will not occur.
  • Values between 0 and 1: Indicate varying degrees of possibility, with numbers closer to 0 being less likely, and those closer to 1 being more likely.
  • 1: Denotes certainty that an event will occur.
  • Negative values and values greater than 1 are not allowed in the probability range.
By adhering to this range, probabilistic predictions remain logical and consistent with real-world observations.
Probability Values Set
Any collection of probability numbers should fall within the defined valid range to be meaningful. When evaluating a set of probabilities, each should be checked whether it adheres to being a part of 0 to 1.
This ensures the completeness and correctness of assessing potential outcomes. If a set contains values outside this scope:
  • Such values could misguide predictions.
  • They do not accurately reflect the likelihood of actual events.
Thus, regularly reviewing probability values sets is essential for ensuring accurate statistical conclusions.
Probability Analysis
Probability analysis involves examining events' chance of occurring, and it forms the basis for decision-making in uncertain situations.
The process begins with identifying valid probabilities and confirming each falls within the permissible range. Analysts evaluate whether each value accurately depicts potential real-world occurrences.
  • Negative values or values beyond 1 are typically eliminated in initial stages.
  • Only those within the 0 to 1 range are further used for successful modeling and predictive analysis.
Having a robust understanding of probability analysis plays a significant role in developing strategies under conditions of uncertainty. By leveraging valid data, one can make well-informed predictions and decisions.

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Most popular questions from this chapter

Find the joint probability of \(A\) and \(B\) for the following. a. \(P(B)=.59\) and \(P(A \mid B)=.77\) b. \(P(A)=.28\) and \(P(B \mid A)=.35\)

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 59 , and then picks a Powerball number from 1 through 39 . The lottery organization randomly draws 5 different white balls from 59 balls numbered 1 through 59 , and then randomly picks a Powerball number from 1 through \(39 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If the player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.) b. If the player's first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about \(\$ 200,000\). Find the probability that a player who buys one ticket will win this prize.

A contractor has submitted bids for two state construction projects. The probability of winning each contract is \(.25\), and it is the same for both contracts a. What is the probability that he will win both contracts? b. What is the probability that he will win neither contract? Draw a tree diagram for this problem.

A small ice cream shop has 10 flavors of ice cream and 5 kinds of toppings for its sundaes. How many different selections of one flavor of ice cream and one kind of topping are possible?

When is the following addition rule used to find the probability of the union of two events \(A\) and \(B\) ? $$P(A \text { or } B)=P(A)+P(B)$$ Give one example where you might use this formula.
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