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Chapter 4: Problem 6

(a) Explain why \(-0.41\) cannot be the probability of some event. (b) Explain why \(1.21\) cannot be the probability of some event. (c) Explain why \(120 \%\) cannot be the probability of some event. (d) Can the number \(0.56\) be the probability of an event? Explain.

Short Answer

Expert verified
(a) -0.41: Cannot be a probability since it's negative. (b) 1.21: Cannot be a probability as it exceeds 1. (c) 120%: Cannot be a probability since it's over 100%. (d) 0.56: Can be a probability as it is within 0 to 1.

Step by step solution

01

Understanding Probability Values

Before we begin, it's important to know that probabilities represent likelihoods that range between 0 and 1 inclusively, where 0 means an impossible event and 1 means a definite event. Additionally, probabilities can be represented as percentages ranging from 0% to 100%.
02

Analyze (a) (-0.41) as Probability

Probabilities must be non-negative, ranging from 0 to 1. Hence, a negative number such as -0.41 cannot represent probability because it falls below 0.
03

Analyze (b) (1.21) as Probability

Probabilities must not exceed 1. The number 1.21 is greater than 1, thus it cannot be a probability as it falls outside the allowable range.
04

Analyze (c) (120%) as Probability

120% is equivalent to the decimal 1.20. As with step 2, probabilities cannot exceed 1, and 120% is greater than 100%, so it cannot be a probability.
05

Analyze (d) (0.56) as Probability

The number 0.56 falls within the interval from 0 to 1. Therefore, 0.56 is an acceptable probability value, as it represents a valid likelihood.

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Key Concepts

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

Probability Range
Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood of an event occurring. One of the key characteristics of probability is its range. The probability range is the set of all possible values that a probability can take. In probability theory:
  • The lowest possible probability is 0, which indicates an event that will not or cannot happen.
  • The highest possible probability is 1, representing an event that is certain to happen.
This range is crucial because it helps us understand whether given values can realistically depict a probability. For instance, if a probability is less than 0 or more than 1, it falls outside this defined range and hence cannot be valid. To visualize it, imagine a number line from 0 to 1; probabilities must reside within this line.
Probability Values
Probability values are the numerical expressions of how likely an event is to occur. These values can be presented in different ways:
  • As decimals, such as 0.25, 0.75, etc.
  • As percentages, like 25% or 75%.
Whether using decimals or percentages, probability values must always adhere to the probability range of 0 to 1, or equivalently from 0% to 100%. This ensures that the probability meaning remains consistent and follows logical principles. A probability value of 1 or 100% means certainty, while a value of 0 or 0% signifies impossibility.
Valid Probabilities
Understanding valid probabilities is essential for accurately modeling and interpreting real-world events. For a probability to be valid:
  • It must be a number between 0 and 1, inclusive.
  • Numbers such as -0.41 or 1.21 do not meet this criterion and are thus invalid as probabilities.
  • Normally, percentages over 100 or numbers less than 0 violate probability rules, making them invalid.
When you say a probability is valid, you mean it's within the limits defined by probability theory. This ensures that any given probability value correctly represents the chance or likelihood of an event.
Probability Interpretation
The interpretation of probability involves understanding what a particular probability value signifies. Each probability reflects a real-world scenario:
  • A probability of 0, or 0%, means the event cannot happen, like drawing an ace of diamonds from a deck with no aces.
  • A probability of 1, or 100%, implies certainty, such as the sun rising in the east.
  • Probabilities between 0 and 1 quantify various degrees of likelihood. For example, a probability of 0.56 suggests a reasonably likely event but not guaranteed.
Interpreting probabilities accurately helps in decision-making and predicting outcomes in uncertain conditions. It requires sensitivity to the nuances between 0% (impossible), 50% (equally likely or unlikely), and 100% (certain).

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Most popular questions from this chapter

You roll two fair dice, a green one and a red one. (a) What is the probability of getting a sum of 6 ? (b) What is the probability of getting a sum of 4 ? (c) What is the probability of getting a sum of 6 or 4? Are these outcomes mutually exclusive?

There are 15 qualified applicants for 5 trainee positions in a fast-food management program. How many different groups of trainees can be selected?

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(\) ace on 1 st card and king on 2 nd). (c) Find \(P(\) king on 1 st card and ace on 2 nd ). (d) Find the probability of drawing an ace and a king in either order.

(a) Draw a tree diagram to display all the possible head-tail sequences that can occur when you flip a coin three times. (b) How many sequences contain exactly two heads? (c) Probability extension: Assuming the sequences are all equally likely, what is the probability that you will get exactly two heads when you toss a coin three times?

The Eastmore Program is a special program to help alcoholics. In the Eastmore Program, an alcoholic lives at home but undergoes a two-phase treatment plan. Phase I is an intensive group-therapy program lasting 10 weeks. Phase II is a long-term counseling program lasting 1 year. Eastmore Programs are located in most major cities, and past data gave the following information based on percentages of success and failure collected over a long period of time: The probability that a client will have a relapse in phase I is \(0.27\); the probability that a client will have a relapse in phase II is 0.23. However, if a client did not have a relapse in phase I, then the probability that this client will not have a relapse in phase II is \(0.95 .\) If a client did have a relapse in phase I, then the probability that this client will have a relapse in phase II is \(0.70\). Let \(A\) be the event that a client has a relapse in phase \(\mathrm{I}\) and \(B\) be the event that a client has a relapse in phase II. Let \(C\) be the event that a client has no relapse in phase I and \(D\) be the event that a client has no relapse in phase II. Compute the following: (a) \(P(A), P(B), P(C)\), and \(P(D)\) (b) \(P(B \mid A)\) and \(P(D \mid C)\) (c) \(P(A\) and \(B)\) and \(P(C\) and \(D)\) (d) \(P(A\) or \(B)\) (e) What is the probability that a client will go through both phase I and phase II without a relapse? (f) What is the probability that a client will have a relapse in both phase I and phase II? (g) What is the probability that a client will have a relapse in either phase I or phase II?
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