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Chapter 7: Problem 29

A die is rolled and the number that falls uppermost is observed. Let \(E\) denote the event that the number shown is a 2 , and let \(F\) denote the event that the number shown is an even number. a. Are the events \(E\) and \(F\) mutually exclusive? b. Are the events \(E\) and \(F\) complementary?

Short Answer

Expert verified
a. No, events \(E\) and \(F\) are not mutually exclusive, as the occurrence of Event \(E\) (rolling a 2) is part of the outcome of Event \(F\) (rolling an even number). b. No, events \(E\) and \(F\) are not complementary, since their occurrence does not imply the non-occurrence of the other event and they do not cover all possible outcomes together.

Step by step solution

01

Define Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time. The occurrence of one event excludes the occurrence of the other event.
02

Define Complementary Events

Complementary events are such that one of these events must occur, and the occurrence of one event implies the non-occurrence of the other event.
03

Evaluate If Events \(E\) and \(F\) are Mutually Exclusive

To determine if events \(E\) and \(F\) are mutually exclusive, we must check if their occurrences are mutually exclusive (meaning the occurrence of one event excludes the occurrence of the other event). Event \(E\) - Number shown is 2 Event \(F\) - Number shown is even number (2, 4, or 6) Since the occurrence of Event \(E\) (rolling a 2) is part of the outcome of Event \(F\) (rolling an even number), these events are not mutually exclusive.
04

Answer Part (a)

Since the events can occur together, they are not mutually exclusive.
05

Evaluate If Events \(E\) and \(F\) are Complementary

To determine if events \(E\) and \(F\) are complementary, we must check if the occurrence of one event implies the non-occurrence of the other event and that they cover all possible outcomes. Event \(E\) - Number shown is 2 Event \(F\) - Number shown is even number (2, 4, or 6) For the events to be complementary, one of the events should cover the outcomes that the other event does not. However, we can observe that event \(F\) also includes the outcome for event \(E\). Thus, events \(E\) and \(F\) cannot be complementary since their occurrence does not imply the non-occurrence of the other event.
06

Answer Part (b)

Since the occurrence of one of the events does not imply the non-occurrence of the other event, they are not complementary events.

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Key Concepts

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

Mutually Exclusive Events
In probability theory, understanding mutually exclusive events is crucial. These are events that cannot occur simultaneously. If one event happens, the other cannot. Think of it like flipping a switch on or off; you can't do both at the same time. For mutually exclusive events, the probability of both events occurring together is zero. When considering two events, say Event A and Event B, they are mutually exclusive if
  • The occurrence of Event A means Event B cannot happen,
  • Mathematically defined as: \( P(A \cap B) = 0 \).

In the exercise, Event E (rolling a 2) and Event F (rolling an even number) were examined. Since rolling a 2 is part of rolling an even number (which also includes 4 and 6), both events can happen together. Therefore, they're not mutually exclusive events.
Complementary Events
Complementary events cover all possible scenarios within a defined sample space. If one event occurs, the complementary event does not, and together they encompass every outcome. They are defined such that: if Event A happens, the complementary Event A' does not. This is expressed as:
  • \( P(A') = 1 - P(A) \)
  • The sum of their probabilities is always equal to 1.

Taking the exercise, for two events to be complementary, like our events E and F:
  • Event E: rolling a 2.
  • Event F: rolling an even number (2, 4, 6).
They cannot be complementary because Event F includes the scenario of Event E. Complementary events should have no overlap, ensuring one covers all outcomes the other does not. Therefore, these two events do not fill the requirement of being complementary.
Dice Rolling Probabilities
Dice rolling exercises offer a practical way to understand probability concepts. A standard die has six faces, each showing a different number from 1 to 6. The probability of any specific outcome is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, giving: \( P( ext{Specific Outcome}) = \frac{1}{6} \).
When analyzing probabilities on dice:
  • Every number 1 through 6 has the same chance, \( P(x) = \frac{1}{6} \).
  • Rolling an even number (2, 4, 6) constitutes three favorable outcomes, thus: \( P(F) = \frac{3}{6} = \frac{1}{2} \).
  • For a specific number like 2, \( P(E) = \frac{1}{6} \).
Using dice rolls to explore mutually exclusive or complementary probabilities is effective as each roll is independent and consistently offers equal opportunity for each outcome. Thus, a practical application in understanding these basic probability principles.

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Most popular questions from this chapter

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or 7 ? Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?

In an attempt to study the leading causes of airline crashes, the following data were compiled from records of airline crashes from 1959 to 1994 (excluding sabotage and military action). $$ \begin{array}{lc} \hline \text { Primary Factor } & \text { Accidents } \\ \hline \text { Pilot } & 327 \\ \hline \text { Airplane } & 49 \\ \hline \text { Maintenance } & 14 \\ \hline \text { Weather } & 22 \\ \hline \text { Airport/air traffic control } & 19 \\ \hline \text { Miscellaneous/other } & 15 \\ \hline \end{array} $$ Assume that you have just learned of an airline crash and that the data give a generally good indication of the causes of airline crashes. Give an estimate of the probability that the primary cause of the crash was due to pilot error or bad weather.

In a poll conducted among likely voters by Zogby International, voters were asked their opinion on the best alternative to oil and coal. The results are as follows: $$ \begin{array}{lcccccc} \hline & & & \text { Fuel } & & \text { Other/ } \\ \text { Source } & \text { Nuclear } & \text { Wind } & \text { cells } & \text { Biofuels } & \text { Solar } & \text { no answer } \\ \hline \text { Respondents, } \% & 14.2 & 16.0 & 3.8 & 24.3 & 27.9 & 13.8 \\ \hline \end{array} $$ What is the probability that a randomly selected participant in the poll mentioned a. Wind or solar energy sources as the best alternative to oil and coal? b. Nuclear or biofuels as the best alternative to oil and coal?

An experiment consists of selecting a card at random from a well-shuffled 52 -card deck. Let \(E\) denote the event that an ace is drawn and let \(F\) denote the event that a spade is drawn. Show that \(n(E \cup F)=n(E)+n(F)-n(E \cap F)\).

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A king of diamonds is drawn.
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