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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

Recent uncertain economic conditions have forced many people to change their spending habits. In a recent telephone poll of 1000 adults, 629 stated that they were cutting back on their daily spending. Suppose that 322 of the 629 people who stated that they were cutting back on their daily spending said that they were cutting back somewhat and 97 stated that they were cutting back somewhat and delaying the purchase of a new car by at least 6 months. If one of the 629 people who are cutting back on their spending is selected at random, what is the probability that he/she is delaying the purchase of a new car by at least 6 months given that he/she is cutting back on spending somewhat?

The probability that a corporation makes charitable contributions is .72. Two corporations are selected at random, and it is noted whether or not they make charitable contributions. Find the probability that at most one corporation makes charitable contributions.

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.

An automated teller machine at a local bank is stocked with \(\$ 10\) and \(\$ 20\) bills. When a customer withdraws \(\$ 40\) from this machine, it dispenses either two \(\$ 20\) bills or four \(\$ 10\) bills. Two customers withdraw \(\$ 40\) each. List all of the outcomes in each of the following events and mention which of these are simple and which are compound events. a. Exactly one customer receives \(\$ 20\) bills. b. Both customers receive \(\$ 10\) bills. c. At most one customer receives \(\$ 20\) bills. d. The first customer receives \(\$ 10\) bills and the second receives \(\$ 20\) bills.

A random sample of 2000 adults showed that 1320 of them have shopped at least once on the Internet. What is the (approximate) probability that a randomly selected adult has shopped on the Internet?
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