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Chapter 12: Problem 2

Probability Says ... Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.) \(\begin{array}{lllllll}0 & 0.05 & 0.45 & 0.50 & 0.55 & 0.95 \quad 1\end{array}\) a. This event is impossible. It can never occur. b. This event is just as likely to occur as it is to not occur. c. This event is very likely, but it will not occur once in a while in a long sequence of trials. d. This event will occur slightly less often than not.

Short Answer

Expert verified
a. 0, b. 0.50, c. 0.95, d. 0.45.

Step by step solution

01

Understand Probability Values

Probability is a measure that indicates the likelihood of an event, ranging from 0 (impossible event) to 1 (certain event). Intermediate values show varying degrees of uncertainty. Here are the values we need to match: - 0 - 0.05 - 0.45 - 0.50 - 0.55 - 0.95 - 1
02

Match Probability for an Impossible Event

An event that is described as impossible means it will never happen. This corresponds to a probability value of 0 because 0 indicates no chance of the event occurring at all.
03

Match Probability for an Even Chance

An event that is just as likely to occur as it is not to occur has a probability of 0.50. It represents an even chance, meaning it could happen or not happen with equal probability.
04

Match Probability for a Very Likely Event

An event that is very likely but might not occur once in a while corresponds to a probability close to, but not exactly, 1. The value 0.95 fits this description, as it suggests the event is almost certain to occur often, but not always.
05

Match Probability for Slightly Less Likely Event

An event that occurs slightly less often than not is expressed by a probability less than 0.50. The value 0.45 indicates such an event, as it is just under the midpoint of 0.50, meaning it happens less frequently.

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Key Concepts

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

Probability Values
Probability values help us quantify how likely an event is to occur. Imagine a scale that goes from 0 to 1.
At 0, events are impossible鈥攖hey simply won't happen. At 1, events are guaranteed and will always happen.
In between these two extremes, we find a spectrum of possibilities.
  • For instance, if the probability is 0.5, it means there's an equal chance of the event happening or not happening.
  • A probability of 0.95, however, means an event is very likely to happen, but there's still a small chance it might not.
  • Conversely, a probability of 0.05 suggests the event is very unlikely.
This numeric representation makes it easier to communicate and calculate the likelihood of different scenarios.
Likelihood
Likelihood refers to how probable it is that a particular event will occur. It's essentially how we talk about probability in everyday language.
Statements like "this event is very likely" or "highly unlikely" translate into numerical probabilities.
  • Very likely could be a probability value such as 0.95, indicating the event will probably happen.
  • Highly unlikely might be 0.05, showing like next to no chance.
Using precise probability values instead of vague terms enables us to describe situations more clearly and make informed decisions based on those numbers.
Events
An event in probability is any outcome or a combination of outcomes from a given scenario.
Events can be anything from rolling a die and getting a six to predicting it's going to rain tomorrow.
  • Each of these possibilities can be assigned a probability value depending on how likely they are to occur, which we call the event's probability.
  • Understanding how to categorize and calculate probabilities for different events is an essential part of working with statistical data.
Being able to recognize and describe events is a foundational skill in probability and statistics.
Uncertainty
Uncertainty is a natural part of probabilistic events. It represents the idea that there's always an element of doubt, no matter how small, regarding the outcome.
In probability, nothing except zero or one is entirely certain.
  • Even if an event is highly likely with a probability of 0.95, there's still a 5% chance it might not happen.
  • Understanding uncertainty helps us make sense of real-life events, as many outcomes can't be predicted with absolute certainty.
By incorporating uncertainty into probability values, we're better equipped to make educated and reasonable predictions.
Statistics
Statistics often employ the concepts of probability to analyze and interpret data.
In the realm of statistics, we use data to discern patterns and anticipate future outcomes, relying heavily on probability.
  • Statistical methods allow us to estimate the probability of various events and understand how data behaves under uncertain conditions.
  • Through statistical analysis, probabilities help us draw conclusions about populations based on sample observations.
Gaining a grasp of how statistics and probability intersect empowers us to tackle complex data problems effectively.

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Most popular questions from this chapter

A Taste Test. A tea-drinking Canadian friend of yours claims to have a very refined palate. She tells you that she can tell if, in preparing a cup of tea, milk is first added to the cup and then hot tea poured into the cup or the hot tea is first poured into the cup and then the milk is added. \(1 .\) To test her claims, you prepare six cups of tea. Three have the milk added first and the other three the tea first. In a blind taste test, your friend tastes all six cups and is asked to identify the three that had the milk added first. a. How many different ways are there to select three of the six cups? (Hint: See Example 12.8, page 281.) b. If your friend is just guessing, what is the probability that she correctly identifies the three cups with the milk added first?

Probability Models? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate- that is, satisfies the rules of probability. Remember, a legitimate model need not be a practically reasonable model. If the assignment of probabilities is not legitimate, give specific reasons for your answer. a. Roll a six-sided die and record the count of spots on the upface: $$ \begin{array}{lll} P(1)=0 & P(2)=1 / 6 & P(3)=1 / 3 \\ P(4)=1 / 3 & P(5)=1 / 6 & P(6)=0 \end{array} $$ b. Deal a card from a shuffled deck: $$ \begin{array}{rlrl} P(\text { clubs }) & =12 / 52 & P(\text { diamonds }) & =12 / 52 \\ P(\text { hearts }) & =12 / 52 & P(\text { spades }) & =16 / 52 \end{array} $$ c. Choose a college student at random and record sex and enrollment status: $$ \begin{array}{rlrl} P(\text { female full-time }) & =0.56 & P(\text { male full-time }) & =0.44 \\\ P(\text { female part-time }) & =0.24 & P(\text { male part-time }) & =0.17 \end{array} $$

Simulating an Opinion Poll. A 2019 Gallup Poll showed that about \(34 \%\) of the American public have very little or no confidence in big business. Suppose that this is exactly true of the population. Choosing a person at random then has probability \(0.34\) of getting one who has very little or no confidence in big business. Use the Probability applet or statistical software to simulate choosing many people at random. (In most software, the key phrase to look for is "Bernoulli trials." This is the technical term for independent trials with Yes/No outcomes. Our outcomes here are "Favorable" or "Not. Favorable.") a. Simulate drawing 50 people, then 100 people, then 400 people. What proportion have very little or no confidence in big business in each case? We expect (but because of chance variation we can't be sure) that the proportion will be closer to \(0.34\) with larger samples. b. Simulate drawing 50 people 10 times and record the percentages in each sample who have very little or no confidence in big business. Then simulate drawing 400 people 10 times and again record the 10 percentages. Which set of 10 results is less variable? We expect the results of samples of size 400 to be more predictable (less variable) than the results of samples of size 50 . That is "long-run regularity" showing itself.

Loaded Dice. There are many ways to produce crooked dice. To load a die so that 6 comes up too often and 1 (which is opposite 6) comes up too seldom, add a bit of lead to the filling of the spot on the 1 face. If a die is loaded so that 6 comes up with probability \(0.2\) and the probabilities of the \(2,3,4\), and 5 faces are not affected, what is the assignment of probabilities to the six faces?

The Medical College Admission Test. The Normal distribution with mean \(\mu=500.9\) and standard deviation \(\sigma=10.6\) is a good description of the total score on the Medical College Admission Test (MCAT). \(\underline{10}\) This is a continuous probability model for the score of a randomly chosen student. Call the score of a randomly chosen student \(X\) for short. a. Write the event "the student chosen has a score of 510 or higher" in terms of \(X\). b. Find the probability of this event.
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