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Chapter 4: Problem 1

Define the following terms: experiment, outcome, sample space, simple event, and compound event.

Short Answer

Expert verified
Experiment in probability is any procedure that can be infinitely repeated and has well-defined outcomes. Outcome is a single result from the set of all possible results. Sample space is the set of all possible outcomes of an experiment. A simple event is an event with exactly one outcome. A compound event consists of more than one simple event.

Step by step solution

01

Define Experiment

In probability, an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of outcomes. Examples include tossing a coin, rolling a dice, drawing a card from a deck, etc.
02

Define Outcome

The outcome of an experiment is a single result from the set of all possible results. For instance, if a coin is tossed, the possible outcomes are either a heads (H) or a tails (T).
03

Define Sample Space

The sample space of an experiment is the set of all possible outcomes. Like for a coin toss, the sample space is {Heads, Tails}. For a roll of a dice, the sample space is {1,2,3,4,5,6}
04

Define Simple Event

A simple event is an event where exactly one outcome can occur at a time. For instance, when a six-sided die is rolled, the possible outcomes 1, 2, 3, 4, 5, and 6 are each a simple event.
05

Define Compound Event

A compound event consists of more than one simple event. It involves the performance of more than one outcome. For example, when two coins are tossed, getting at least one head is a compound event since there are several ways it can occur (HT, TH, HH).

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Key Concepts

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

Experiment
In probability, an experiment is a process or action that can be repeated numerous times, resulting in a set of outcomes known as the experiment's possible outcomes. Experiments are pivotal as they form the basis for statistical analysis and probability theory. They are typically consistent and repeatable, meaning that the conditions under which the experiment is conducted remain the same each time. Examples of common probability experiments include:
  • Tossing a coin
  • Rolling a dice
  • Drawing a card from a deck
During these experiments, the outcome will vary, but the process remains unchanged.
Outcome
An outcome is a possible result that stems from conducting an experiment. Each time an experiment is performed, one particular result will occur, representing the completion of one trial of the experiment. For example, consider the act of flipping a coin: there are two possible outcomes here, heads or tails. Each side represents a single, distinct outcome. Outcomes can vary in complexity depending on the type of experiment but are generally specific and measurable results.
Sample Space
The sample space is a fundamental concept in probability, representing the complete set of all possible outcomes of an experiment. Knowing the sample space is crucial as it allows us to comprehend the range of possible events that could occur. For instance:
  • Tossing a single coin has a sample space of {Heads, Tails}.
  • Rolling a six-sided dice yields a sample space of {1, 2, 3, 4, 5, 6}.
Each potential result in an experiment's sample space is unique, ensuring that every possibility is accounted for when evaluating probabilities. Understanding the sample space helps in identifying all possible outcomes, which is essential for calculating the likelihood of various events.
Simple Event
A simple event in probability concerns the occurrence of exactly one outcome from a sample space. Simple events are the most basic type of events and are often used as building blocks for more complex analyses. For instance, if a single die is rolled, landing on a 3 is a simple event since it represents just one of the outcomes from the complete sample space of {1, 2, 3, 4, 5, 6}. Since only one result or occurrence is involved, it simplifies the process of calculation and understanding of basic probability concepts.
Compound Event
Contrasting with a simple event, a compound event involves the occurrence of two or more outcomes from an experiment. Compound events are more complex and can be the result of performing multiple experiments or conducting a single experiment that produces several outcomes. For example, consider the scenario of tossing two coins simultaneously:
  • Getting at least one head involves the compound events HH, HT, and TH, meaning multiple outcomes contribute to the event being realized.
  • Rolling a dice and getting an even number could include the compound events {2, 4, 6}.
Compound events are crucial for understanding how multiple outcomes can interact and the overall probability of various combinations occurring during experiments.

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Most popular questions from this chapter

A player plays a roulette game in a casino by betting on a single number each time. Because the wheel has 38 numbers, the probability that the player will win in a single play is \(1 / 38\). Note that each play of the game is independent of all previous plays. a. Find the probability that the player will win for the first time on the 10 th play. b. Find the probability that it takes the player more than 50 plays to win for the first time. c. A gambler claims that because he has 1 chance in 38 of winning each time he plays, he is certain to win at least once if he plays 38 times. Does this sound reasonable to you? Find the probability that he will win at least once in 38 plays.

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is .13. If the patient does not have the disease, the probability that the test indicates a (false) positive is 10 . Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive

A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to \(3599 .\) To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

In a large city, 15,000 workers lost their jobs last year. Of them, 7400 lost their jobs because their companies closed down or moved, 4600 lost their jobs due to insufficient work, and the remainder lost their jobs because their positions were abolished. If one of these 15,000 workers is selected at random, find the probability that this worker lost his or her job a. because the company closed down or moved b. due to insufficient work c. because the position was abolished Do these probabilities add to \(1.0 ?\) If so, why?

Given that \(P(B)=.29\) and \(P(A\) and \(B)=.24\), find \(P(A \mid B)\).
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