/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Let \(A\) be the event that a nu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 63

Let \(A\) be the event that a number less than 3 is obtained if we roll a die once. What is the probability of \(A ?\) What is the complementary event of \(A\), and what is its probability?

Short Answer

Expert verified
The probability of event \(A\) is \(\frac{1}{3}\) and the probability of the complementary event \(A'\) is \(\frac{2}{3}\).

Step by step solution

01

Calculate Total Possible Outcomes

A die has six faces. Each roll of a die is independent and can result in any one of these six outcomes. Therefore, the total number of possible outcomes when rolling a die once is 6.
02

Identify Favorable Outcomes For Event \(A\)

Event \(A\), as defined in the exercise, is obtaining a number that's less than 3 when rolling a die. The outputs that satisfy this condition are 1 and 2. So, there are 2 favorable outcomes for event \(A\).
03

Calculate Probability of Event \(A\)

The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. Therefore, the probability of event \(A\) is \(\frac{2}{6} = \frac{1}{3}\).
04

Identify Complementary Event

The complementary event of \(A\), denoted by \(A'\), is the event that \(A\) does not occur. This means if we roll a die, we get a number that is not less than 3. The numbers that fulfill this condition are 3, 4, 5, and 6. Hence, there are 4 favorable outcomes for event \(A'\).
05

Calculate Probability of Complementary Event \(A'\)

The probability of event \(A'\) is calculated the same way as the probability of event \(A\). It's the ratio of the number of favorable outcomes for \(A'\) to the total number of possible outcomes. Hence, the probability of \(A'\) is \(\frac{4}{6} = \frac{2}{3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Complementary Events
In probability, complementary events are a fascinating concept. When we talk about an event \(A\), its complement, \(A'\), is the scenario where \(A\) doesn't happen. It covers all other possible outcomes.
This way, if the probability of event \(A\) is known, the probability of \(A'\) is just what's left over.
  • The formula for the complementary event is: \( P(A') = 1 - P(A) \)
So, if rolling a die yields a number less than 3 (event \(A\)), the complementary event \(A'\) gives a number 3 or greater. This simple relationship helps solve many probability problems easily.
Favorable Outcomes
Understanding favorable outcomes is key in probability. When you're given an event, favorable outcomes are those specific results that lead to the event happening.
  • For event \(A\), these are the numbers less than 3: 1 and 2.
  • Thus, there are 2 favorable outcomes for \(A\).
A crucial concept is counting these outcomes correctly and determining their probability based on the total possible outcomes. This forms the foundation for calculating event likelihood.
Rolling a Die
Rolling a die is a classic example in probability, often used because of its simplicity and predictability. A standard die has 6 faces, numbered from 1 to 6, making it a great tool for learning.
Each roll of a die is a random, independent trial with six possible outcomes.
  • Each face has an equal probability of \(\frac{1}{6}\).
  • Rolling provides a clear method to visualize total and favorable outcomes.
This experiment also introduces concepts like fairness and independence, making it a staple in probability education.
Independent Events
Independent events in probability are outcomes where one event does not impact the other. Each die roll stands alone, not affecting any subsequent rolls.
For example, rolling a 2 now doesn鈥檛 influence what number shows up next.
  • Mathematically, two events \(A\) and \(B\) are independent if \( P(A \cap B) = P(A) P(B) \).
  • This means knowing one doesn't inform about the other.
Understanding this helps in more complex scenarios, where multiple events occur without affecting each other鈥檚 outcomes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given that \(P(A)=.30\) and \(P(A\) and \(B)=.24\), find \(P(B \mid A)\).

A test contains two multiple-choice questions. If a student makes a random guess to answer each question, how many outcomes are possible? Depict all these outcomes in a Venn diagram. Also draw a tree diagram for this experiment. (Hint: Consider two outcomes for each question- either the answer is correct or it is wrong.)

What is meant by two mutually exclusive events? Give one example of two mutually exclusive events and another example of two mutually nonexclusive events.

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 59 , and then picks a Powerball number from 1 through 39 . The lottery organization randomly draws 5 different white balls from 59 balls numbered 1 through 59 , and then randomly picks a Powerball number from 1 through \(39 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If the player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.) b. If the player's first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about \(\$ 200,000\). Find the probability that a player who buys one ticket will win this prize.

Given that \(A\) and \(B\) are two independent events, find their joint probability for the following. a. \(P(A)=.61\) and \(P(B)=.27\) b. \(P(A)=.39\) and \(P(B)=.63\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.