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Chapter 4: Problem 3

What is the probability of (a) an event \(A\) that is certain to occur? (b) an event \(B\) that is impossible?

Short Answer

Expert verified
(a) The probability of event A is 1. (b) The probability of event B is 0.

Step by step solution

01

Understanding Probability

Probability is a measure of how likely an event is to occur. It ranges between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.
02

Probability of a Certain Event

For event \(A\), which is certain to occur, the probability is the highest possible value because there is no doubt about its occurrence. Thus, \(P(A) = 1\).
03

Probability of an Impossible Event

For event \(B\), which is impossible, the probability is the lowest possible value because there is no chance of occurrence. Therefore, \(P(B) = 0\).

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Key Concepts

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

Certain Event
A certain event in probability theory is one that will definitely occur when an experiment is conducted. This type of event has no uncertainty associated with it, meaning its occurrence is guaranteed. To better understand, imagine flipping a two-sided coin you know will never land on its edge. The event of the coin landing either heads or tails is certain. In mathematical terms, this certainty translates to a probability value of 1, indicating 100% likelihood.
  • Example of a certain event is the sun rising in the east every day.
  • Rolling any number between 1 to 6 on a standard die guarantees at least one number is rolled.
In conclusion, certain events are absolute in probability, rooting from the belief that there are no other possible outcomes once the event is guaranteed to occur.
Impossible Event
An impossible event is one that will never happen, no matter how many times the experiment is repeated or how long you wait. The aspect of impossibility in these events is quite intuitive. Consider an example where one is asked to pick a red ball from a bag containing only green balls. The chance of picking a red ball is zero, and this zero probability reflects the impossibility.
  • If you were asked to find the probability of selecting a day of the week that is not Monday among a set containing only Monday, this would be an impossible event.
  • Another example is rolling a 7 with a standard six-sided die.
Impossible events are assigned a probability value of 0 within probability theory itself and represent situations where no conceivable effort can lead to the occurrence of the event.
Probability Values
The concept of probability values provides a numerical way to express the likelihood of events. Probability values range from 0 to 1, where:
  • A probability of 0 represents an impossible event.
  • A probability of 1 represents a certain event.
  • A probability between 0 and 1 represents various degrees of likelihood for events that are neither certain nor impossible.
Moreover, probability values help us assess risk, make predictions, and understand phenomena based on the frequency or logic. For instance, a probability value of 0.5, or 50%, might represent the chance of a single coin flip landing heads.
Probability finds its foundation in real-world scenarios such as games of chance, weather forecasting, and statistical modeling. By effectively using probability values, we enhance our decision-making skills in uncertain situations.

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Most popular questions from this chapter

(a) Make a tree diagram to show all the possible sequences of answers for three multiple-choice questions, each with four possible responses. (b) Probability Extension Assuming that you are guessing the answers so that all outcomes listed in the tree are equally likely, what is the probability that you will guess the one sequence that contains all three correct answers?

(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?

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If two events are mutually exclusive, can they occur concurrently? Explain.

Consider the following events for a college student selected at random: \(A=\) student is female \(B=\) student is majoring in business Translate each of the following phrases into symbols. (a) The probability the student is male or is majoring in business (b) The probability a female student is majoring in business (c) The probability a business major is female (d) The probability the student is female and is not majoring in business (e) The probability the student is female and is majoring in business
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