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Chapter 4: Problem 46

A statistical experiment has 10 equally likely outcomes that are denoted by \(1,2,3,4,5,6,7,8,9\), and \(10 .\) Let event \(A=\\{3,4,6,9\\}\) and event \(B=\\{1,2,5\\}\). 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?

Short Answer

Expert verified
a. Yes, events A and B are mutually exclusive. b. Yes, events A and B are independent. c. The complements of events A and B are \(A' = \{1,2,5,7,8,10\}\) and \(B' = \{3,4,6,7,8,9,10\}\) respectively, with their probabilities being 0.6 and 0.7 respectively.

Step by step solution

01

Check for mutual exclusivity

The sets representing events A and B have no elements in common, as A = \{3,4,6,9\} and B = \{1,2,5\}. Therefore, events A and B are mutually exclusive.
02

Check for independence

Two events are independent if the probability of one event occurring does not change the probability of the other event. Since events A and B do not share any outcomes, the occurrence of one event does not affect the occurrence of the other. Hence, events A and B are independent.
03

Find the complements and their probabilities

The complement of an event is the set of all outcomes that are not in the given event. Therefore, \(A' = \{1,2,5,7,8,10\}\) and \(B' = \{3,4,6,7,8,9,10\}\). As there are ten equally likely outcomes, \(P(A') = \frac{6}{10} = 0.6\) and \(P(B') = \frac{7}{10} = 0.7\).

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Key Concepts

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

Mutually Exclusive Events
When we talk about mutually exclusive events in probability theory, we are referring to events that cannot occur simultaneously. In other words, if one event happens, the other cannot.
In the context of the exercise, events \(A\) and \(B\) are represented as \(A = \{3,4,6,9\}\) and \(B = \{1,2,5\}\).
Clearly, these two sets have no outcomes in common.
  • If event \(A\) occurs, the outcome is either 3, 4, 6, or 9.
  • If event \(B\) happens, the result must be 1, 2, or 5.
Since none of these numbers overlap, \(A\) and \(B\) can never be true at the same time. This means they are indeed mutually exclusive.
Independent Events
Independent events take on a different kind of relationship in probability theory than mutually exclusive events. Here, the occurrence of one event does not influence the probability of the other occurring.
In our exercise, events \(A = \{3,4,6,9\}\) and \(B = \{1,2,5\}\) are independent, because the outcome of one event does not affect the outcomes of the other.
  • This means that knowing the result of event \(A\) gives us no information about the result of event \(B\), and vice versa.
  • Additionally, independence implies that the probability of both \(A\) and \(B\) occurring together equals the product of their individual probabilities, though for mutually exclusive events this probability is zero.
Complement of an Event
The complement of an event includes all outcomes that are not part of the event. This is a vital concept in probability theory because it provides another way of calculating probabilities.
In the exercise, the complement of event \(A\) (\(A'\)) is all the outcomes not in \(A\):
  • \(A' = \{1,2,5,7,8,10\}\)
Similarly, the complement of \(B\) (\(B'\)) consists of:
  • \(B' = \{3,4,6,7,8,9,10\}\)
Since there are 10 equally likely outcomes and both events detail which are excluded and included, calculating the probabilities of the complements becomes straightforward:
  • \(P(A') = \frac{6}{10} = 0.6\)
  • \(P(B') = \frac{7}{10} = 0.7\)
Equal Likelihood
Equal likelihood refers to a situation in which all possible outcomes of an experiment have the same probability of occurring. It is often assumed in textbook exercises to simplify calculations and focus on understanding core probability concepts.
In the given exercise, each of the 10 outcomes (1 through 10) is considered equally probable.
  • This means each individual event has a probability of \(\frac{1}{10}\).
Understanding this is crucial for calculating overall probabilities and realizing why complements and other events behave as they do in this scenario.
Probability Calculation
Probability calculation is the process of determining the likelihood of a given event occurring in an experiment or random trial.
To find the probability of an event, you can divide the number of favorable outcomes by the total number of possible outcomes.
  • For event \(A = \{3,4,6,9\}\), the probability \(P(A)\) is \(\frac{4}{10} = 0.4\).
  • For event \(B = \{1,2,5\}\), the probability \(P(B)\) is \(\frac{3}{10} = 0.3\).
This basic principle underlies all probability calculations and is essential for understanding more advanced concepts in probability theory.

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Most popular questions from this chapter

Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses obtained. $$ \begin{array}{lcc} \hline & \text { Have Shopped } & \text { Have Never Shopped } \\ \hline \text { Male } & 500 & 700 \\ \text { Female } & 300 & 500 \\ \hline \end{array} $$ a. Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. i. \(P\) (has never shopped on the Internet and is a male) ii. \(P\) (has shopped on the Internet and is a female) b. Mention what other joint probabilities you can calculate for this table and then find those. You may draw a tree diagram to find these probabilities.

Refer to Exercise 4.48. A 2010-2011 poll conducted by Gallup (www.gallup.com/poll/148994/ Emotional-Health-Higher-Among-Older- Americans.aspx) examined the emotional health of a large number of Americans. Among other things, Gallup reported on whether people had Emotional Health Index scores of 90 or higher, which would classify them as being emotionally well-off. The report was based on a survey of 65,528 people in the age group \(35-44\) years and 91,802 people in the age group \(65-74\) years. The following table gives the results of the survey, converting percentages to frequencies. $$ \begin{array}{lcc} \hline & \text { Emotionally Well-Off } & \text { Emotionally Not Well-Off } \\\ \hline \text { 35-44 Age group } & 16,016 & 49,512 \\ \text { 65-74 Age group } & 32,583 & 59,219 \\ \hline \end{array} $$ a. Suppose that one person is selected at random from this sample of 157,330 Americans. Find the following probabilities. i. \(P(35-44\) age group and emotionally not well-off \()\) ii. \(P(\) emotionally well-off and \(65-74\) age group \()\) b. Find the joint probability of the events \(35-44\) age group and \(65-74\) age group. Is this probability zero? Explain why or why not.

Amy is trying to purchase concert tickets online for two of her favorite bands, the Leather Recliners and Double Latte No Foam. She estimates that her probability of being able to get tickets for the Leather Recliners concert is .14, the probability of being able to get tickets for the Double Latte No Foam concert is \(.23\), and the probability of being able to get tickets for both concerts is \(.026 .\) What is the probability that she will be able to get tickets for at least one of the two concerts?

Suppose that \(20 \%\) of all adults in a small town live alone, and \(8 \%\) of the adults live alone and have at least one pet. What is the probability that a randomly selected adult from this town has at least one pet given that this adult lives alone?

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