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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 \(\frac{4}{6} = \frac{2}{3}\).

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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.

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Most popular questions from this chapter

A company is to hire two new employees. They have prepared a final list of eight candidates, all of whom are equally qualified. Of these eight candidates, five are women. If the company decides to select two persons randomly from these eight candidates, what is the probability that both of them are women? Draw a tree diagram for this problem.

In a sample survey, 1800 senior citizens were asked whether or not they have ever been victimized by a dishonest telemarketer. The following table gives the responses by age group. $$\begin{array}{l|llcc} & & & \begin{array}{c} \text { Have Been } \\ \text { Victimized } \end{array} & \begin{array}{c} \text { Have Never } \\ \text { Been Victimized } \end{array} \\ \hline & 60-69 & \text { (A) } & 106 & 698 \\ \text { Age } & 70-79 & \text { (B) } & 145 & 447 \\ & 80 \text { or over (C) } & 61 & 343 \\ \hline \end{array}$$ Suppose one person is randomly selected from these senior citizens. Find the following probabilities a. \(P(\) have been victimized or \(\mathrm{B}\) ) b. \(P(\) have never been victimized or \(\mathrm{C}\) )

A production system has two production lines; each production line performs a two-part process, and each process is completed by a different machine. Thus, there are four machines, which we can identify as two first-level machines and two second-level machines. Each of the first-level machines works properly \(98 \%\) of the time, and each of the second-level machines works properly \(96 \%\) of the time. All four machines are independent in regard to working properly or breaking down. Two products enter this production system, one in each production line a. Find the probability that both products successfully complete the two-part process (i.e., all four machines are working properly). b. Find the probability that neither product successfully completes the two- part process (i.e., at least one of the machines in each production line is not working properly).

A company has installed a generator to back up the power in case there is a power failure. The probability that there will be a power failure during a snowstorm is \(.30\). The probability that the generator will stop working during a snowstorm is .09. What is the probability that during a snowstorm the company will lose both sources of power? Note that the two sources of power are independent.

A statistical experiment has eight equally likely outcomes that are denoted by \(1,2,3,4,5,6,7\), and 8\. Let event \(A=\\{2,5,7\\}\) and event \(B=\\{2,4,8\\}\). a. Are events \(A\) and \(B\) mutually exclusive events? b. Are events \(A\) and \(B\) independent events? c. What are the complements of events \(A\) and \(B\), respectively, and their probabilities?
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