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Chapter 5: Problem 4

What does it mean when two events are complements? In Problems \(5-12,\) a probability experiment is conducted in which the sample space of the experiment is \(S=\\{1,2,3,4,5,6,7,8,9,\) 10,11,12\\}\(.\) Let event \(E=\\{2,3,4,5,6,7\\},\) event \(F=\\{5,6,7,8,9\\},\) event \(G=\\{9,10,11,12\\},\) and event \(H=\\{2,3,4\\} .\) Assume that each outcome is equally likely.

Short Answer

Expert verified
Two events are complements if they together form the entire sample space without overlapping.

Step by step solution

01

Define Complements

Two events are complements if one event occurs if and only if the other event does not occur. In other words, event A and event A' (A complement) make up the entire sample space S, and they do not overlap.
02

Identify the Sample Space

The total sample space is given as \[ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \]Each outcome is equally likely.
03

Define Events

The events given are: Event \[ E = \{2, 3, 4, 5, 6, 7\} \]Event \[ F = \{5, 6, 7, 8, 9\} \]Event \[ G = \{9, 10, 11, 12\} \]Event \[ H = \{2, 3, 4\} \]
04

Find Complements

For each event, identify their complements. The complement of event E, represented as E', is all elements in S that are not in E:\[ E' = S - E = \{1, 8, 9, 10, 11, 12\} \]Similarly, \[ F' = \{1, 2, 3, 4, 10, 11, 12\} \]\[ G' = \{1, 2, 3, 4, 5, 6, 7, 8\} \]\[ H' = \{1, 5, 6, 7, 8, 9, 10, 11, 12\} \]
05

Cross-Check Complements

To ensure correctness, ensure that each event and its complement add up to the total sample space without overlapping:\[ E + E' = S \]\[ F + F' = S \]\[ G + G' = S \]\[ H + H' = S \]

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Key Concepts

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

Sample Space
In probability theory, the sample space is the set of all possible outcomes of a probability experiment. Think of it as the 'universe' of your experiment. For example, if you're rolling a 6-sided die, your sample space would be:

\[ S = \{1, 2, 3, 4, 5, 6\} \].
  • Each number represents a possible outcome.
  • No outcome is left out, covering every possibility.
In our specific exercise, the sample space was given as:
\[ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \]
This means there are 12 possible outcomes when you conduct the probability experiment (like maybe rolling a 12-sided die). Ensure whenever you talk about events, they fit within the sample space.
Probability Experiment
A probability experiment is any action or process that leads to one or several outcomes. Examples include flipping a coin, rolling a die, or drawing a card from a deck. These experiments form the basis for probability calculations.
  • When you flip a coin, the experiment generates one of two outcomes: Heads or Tails.
  • When you roll a die, the experiment generates an outcome from the die鈥檚 faces.
In our exercise, performing the action (experiment) results in one of the 12 outcomes in our sample space:
\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \]
Understanding what constitutes these experiments will help you comprehend further probability concepts accurately.
Equally Likely Outcomes
For an accurate probability analysis, it is crucial to know if outcomes are equally likely. When we say outcomes are 'equally likely,' it means each outcome has the same probability of occurring.
  • In a fair 6-sided die, numbers 1 through 6 are equally likely; each has a chance of \( \frac{1}{6} \).
  • In a fair coin flip, both heads and tails have a chance of \( \frac{1}{2} \).
In our exercise, the outcomes in the sample space:
\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \]
are stated to be equally likely. This assumption is vital as it allows us to treat all 12 possibilities with the same weight in probability calculations.
Event Complements
Two events are complements if they are mutually exclusive and together cover the entire sample space. If Event A occurs, Event A' (the complement) does not occur, and vice versa. For example, if Event A = rolling an even number on a die, Event A' = rolling an odd number.
  • To find A's complement, we exclude all the outcomes that are in A from the sample space.
  • Event and its complement should add up to the total sample space.
In our exercise:
\[ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \]
Given Event \(E = \{2, 3, 4, 5, 6, 7\} \), its complement \(E'\) is:
\[ E' = \{1, 8, 9, 10, 11, 12\} \]
Always ensure you triple-check that combining these sets gives back the original sample space with no overlaps or gaps.

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Most popular questions from this chapter

In Problems 13-18, find the probability of the indicated event if \(P(E)=0.25\) and \(P(F)=0.45\) \(P\left(F^{c}\right)\)

You are dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52-card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

Suppose that Ralph gets a strike when bowling \(30 \%\) of the time. (a) What is the probability that Ralph gets two strikes in a row? (b) What is the probability that Ralph gets a turkey (three strikes in a row)? (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Ralph gets a turkey, but fails to get a clover (four strikes in a row).

The probability that a randomly selected individual in the United States 25 years and older has at least a bachelor's degree is \(0.094 .\) The probability that an individual in the United States 25 years and older has at least a bachelor's degree, given that the individual lives in Washington \(\mathrm{DC},\) is, \(0.241 .\) Are the events "bachelor's degree" and "lives in Washington, DC," independent?

List all the permutations of four objects \(a, b, c,\) and \(d\) taken two at a time without repetition. What is \({ }_{4} P_{2} ?\)
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