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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.) $$ \begin{array}{rrrrrrrr} 00.010 .450 .500 .550 .9910 & 0.01 & 0.45 & 0.50 & 0.55 & 0.99 & 1 \end{array} $$ (a) This event is impossible. It can never occur. (b) This event is certain. It will occur on every trial. (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) 0.01, b) 1, c) 0.99, d) 0.45.

Step by step solution

01

Understanding Probabilities

Probabilities range from 0 to 1. A probability of 0 means the event is impossible, whereas a probability of 1 means the event is certain to occur.
02

Match the Impossible Event

The statement 'This event is impossible. It can never occur' corresponds to a probability of 0. So the probability to match here is 0.01.
03

Match the Certain Event

The statement 'This event is certain. It will occur on every trial' corresponds to a probability of 1.
04

Match the Very Likely Event

The statement 'This event is very likely, but it will not occur once in a while in a long sequence of trials' suggests a probability very close to 1 (like 0.99), acknowledging there is still a small chance it might not occur.
05

Match the Slightly Less Likely Event

The statement 'This event will occur slightly less often than not' refers to a probability slightly less than 0.50. Thus, match it with 0.45.

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Key Concepts

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

Event Likelihood
Probability is a helpful tool that gives us a way to talk about how likely something is to happen. Event likelihood helps us understand the chance of an event occurring in a practical, numerical way.
For instance, when we say an event is likely, we are referring to a probability that leans toward the higher end of the scale. If someone says there's a 70% chance of rain, they mean it is more probable than not.
Conversely, if we say an event is unlikely, we reference a lower percentage, showing that it is less likely to occur. A probability of 10% suggests something probably won't happen.
  • A probability closer to 0 means the event is unlikely or rare.
  • A probability closer to 1 suggests an event is likely or expected.
This understanding of likelihood allows us to make better predictions and decisions daily.
Probability Range
Probability is always measured between 0 and 1. This range helps us define the spectrum of certainty for any event. Imagine a line where 0 sits on one end and 1 on the other. Each point on this line represents a different degree of likelihood.
The probability range, therefore, acts as a sort of anchor that binds our expectations in a predictable pattern.
This makes it easy for people to talk about uncertainty with a common language.
  • 0 represents an impossible event.
  • 1 represents an event that is certain.
  • Any number in between reflects how likely it is for an event to happen.
Remember, probability does not always guarantee an outcome but gives us strong indications.
Certainty in Probability
Certainty in probability is expressed with the number 1. When an event has a probability of 1, it means that event is going to happen every single time under given circumstances. It's like the sun rising each morning or water being wet.
Events this certain are rare because most things in life come with some level of unpredictability. However, when you find that elusive event with a probability of 1, you can be as confident as possible that it will occur on every trial.
  • An event with a probability of 1 demonstrates full confidence.
  • This degree of certainty is mostly theoretical with few practical examples.
Understanding this helps in distinguishing between assumptions and facts.
Impossibility in Probability
Understanding impossibility in probability is simple once you know that a probability of 0 means an event will not occur. Think of it like trying to walk through a solid wall without magic鈥攊t just doesn't happen.
Assigning a probability of 0 to an event communicates that, given the current circumstances, it is impossible for this outcome to occur.
  • Probability 0 is absolute in showing impossibility.
  • If conditions do not change, the event remains impossible.
This concept helps in eliminating outcomes that have no chance of happening, allowing focus on more probable expectations.

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Most popular questions from this chapter

Who gets interviewed? Abby, Deborah, Mei-Ling, Sam, and Roberto are students in a small seminar course. Their professor decides to choose two of them to interview about the course. To avoid unfairness, the choice will be made by drawing two names from a hat. (This is an SRS of size 2.) (a) Write down all possible choices of two of the five names. This is the sample space. (b) The random drawing makes all choices equally likely. What is the probability of each choice? (c) What is the probability that Mei-Ling is chosen? (d) Abby, Deborah, and Mei-Ling liked the course. Sam and Roberto did not like the course. What is the probability that both people selected liked the course?

Are They Disjoint? Which of the following pairs of events, \(A\) and \(B\), are disjoint? Explain your answers. (a) A person is selected at random. A is the event \({ }^{\alpha}\) sex of the person selected is male"; \(B\) is the event "sex of the person selected is female." (b) A person is selected at random. \(A\) is the event "the person selected earns more than \(\$ 100,000\) per year"; \(B\) is the event "the person selected earns more than \(\$ 250,000\) per year." (c) A pair of dice are tossed. \(A\) is the event "one of the dice is a \(3^{\prime \prime}, B\) is the event "the sum of the two dice is 3 ."

1\. Choose a common fruit fly Drosophila melanogaster at random. Call the length of the thorax (where the wings and legs attach) \(Y\). The random variable \(Y\) has the Normal distribution with mean \(\mu=0.800\) millimeter (mm) and standard deviation \(\sigma=0.078 \mathrm{~mm}\). The probability \(P(Y>1)\) that the fly you choose has a thorax more than \(1 \mathrm{~mm}\) long is about (a) \(0.995\) (b) \(0.5\) (c) \(0.005\).

Education among young adults. Choose a young adult (aged 25-29) at random. The probability is \(0.10\) that the person chosen did not complete high school, \(0.27\) that the person has a high school diploma but no further education, and \(0.34\) that the person has at least a bachelor's degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor's degree? (b) What is the probability that a randomly chosen young adult has at least a high school education?

Nickels falling over. You may feel that it is obvious that the probability of a head in tossing a coin is about \(1 / 2\) because the coin has two faces. Such opinions are not always correct. Stand a nickel on edge on a hard, flat surface. Pound the surface with your hand so that the nickel falls over. What is the probability that it falls with heads upward? Make at least 50 trials to estimate the probability of a head.
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