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Chapter 16: Problem 12

A student randomly guesses the answers to a four-question true-or-false quiz. The observation is the student's answer (T or \(F\) ) for each question [see Exercise \(2(\mathrm{~b})\) ]. Write out the event described by each of the following statements as a set. (a) \(E_{1}:\) "the student answers \(T\) to two out of the four questions." (b) \(E_{2}\) : "the student answers \(T\) to at least two out of the four questions." (c) \(E_{3}\) : "the student answers \(T\) to at most two out of the four questions." (d) \(E_{4}:\) "the student answers \(T\) to the first two questions."

Short Answer

Expert verified
From the analysis, the sets for events \(E_{1}\), \(E_{2}\), \(E_{3}\), and \(E_{4}\) comprise the following elements: \(E_{1} = \){'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT'}, \(E_{2} = \){'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT', 'TTTF', 'TFFT', 'FTTT', 'FFTT', 'TFFF', 'TTFF', 'TTT', 'TTTT'}, \(E_{3} = \){'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT', 'TF', 'FT', 'TT', 'FFFF'}, and \(E_{4} = \){'TTTT', 'TTTF', 'TTFT', 'TTFF'} respectively.

Step by step solution

01

Solution for Part (a)

Here are all possible orders for the student answering True to two out of four questions: TFTF, TTFF, FTTF, FFTT, TFFT, FTT. Any other combination would result in more or less than 2 True answers. Thus, the event \(E_1\) can be defined as the set \(E_1 = \){'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT' }.
02

Solution for Part (b)

For the student answering True to at least two out of four questions, the possible combinations are the same as in part a, plus those where the student answers True to three or four questions. So \(E_2 = \){'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT', 'TTTF', 'TFFT', 'FTTT', 'FFTT', 'TFFF', 'TTFF', 'TTT', 'TTTT'}.
03

Solution for Part (c)

For the student answering True to at most two out of four questions, the possible combinations are the same as in part a, plus those where the student answers True to one or zero questions. So \(E_3 = \){'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT', 'TF', 'FT', 'TT', 'FFFF'}.
04

Solution for Part (d)

For the student answering True to the first two questions, the remaining two can be either True or False in any combination. So, the combinations are TTTT, TTTF, TTFT, and TTFF. Thus, \(E_4 = \){'TTTT', 'TTTF', 'TTFT', 'TTFF'}.

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics that focuses on counting, arranging, and finding patterns. It is a powerful tool used to determine the number of possible outcomes in various scenarios. For a quiz consisting of true-or-false questions, combinatorics helps in understanding all the ways a student can choose answers.
Each question provides two possible answers: True (T) or False (F). Combinatorial methods count all possible combinations and arrangements of these outcomes across multiple questions.
  • Consider a quiz with four questions. Each question can be independently answered with T or F.
  • The number of possible answer combinations is determined by multiplying the choices for each question: \(2^4 = 16\)
Understanding combinatorics assists students in visualizing all possible arrangements, making it easier to solve complex probability problems.
True-or-False Quiz
A true-or-false quiz provides a simple binary system of answering, where for each question only two options are available: True or False. This simplicity makes it a perfect model for exploring basic probability and combinatorial concepts.
In terms of probability:
  • Each question has an equal probability of being answered as True or False, assuming guessing.
  • For a single question, the probability is \(\frac{1}{2}\) for True and \(\frac{1}{2}\) for False.

When dealing with multiple questions, combinations of answers create diverse sets for event analysis. Such quizzes exemplify how simple scenarios can illustrate fundamental mathematical principles in probability.
Event Outcomes
Event outcomes in probability are the different results that one can expect from a particular experiment or action. Each outcome is one possible result that can occur during the event. In the context of a true-or-false quiz:
  • Every specific sequence of answers (e.g., 'TFTF') is an individual outcome.
  • Outcomes can be grouped to define various events like getting exactly two True answers.
Analyzing the outcome involves listing all the combinations of T's and F's across the quiz questions to form a comprehensive set of possible results. Once all potential outcomes are considered, specific events of interest are identified, such as those that meet certain criteria (e.g., two or three T's). Employing set notation helps encapsulate these outcomes in a structured manner for easy analysis and interpretation.
Set Notation
Set notation is a mathematical way to define a collection of distinct objects, considered as a unit. In probability, set notation helps in organizing and expressing the different possible outcomes of an event. For the true-or-false quiz question example:
  • If we want to note down all the ways a student can answer exactly two True questions out of four, we use set notation like \(E_1 = \{ 'TFTF', 'TTFF', 'FTTF', 'FFTT', 'TFFT', 'FTT' \}\).
  • This method allows grouping outcomes that meet a specific criterion, and clearly represents them for solving probability questions.
Set notation is crucial in simplifying and organizing complex outcomes, allowing students to better comprehend and tackle the various possibilities in any probability task.

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Most popular questions from this chapter

Ten professional basketball teams are participating in a draft lottery. (A draft lottery is a lottery to determine the order in which teams get to draft players.) Ten balls, each containing the name of one team (call them \(A, B, C, D, E\), \(F, G, H, I,\) and \(J\) for short), are placed in an urn and thoroughly mixed. Four balls are drawn, one at a time, from the urn. The four teams chosen get to draft first and in the order they are chosen. The remaining six teams have to draft in reverse order of season records. Find the probability that (a) \(A\) is the first team chosen. (b) \(A\) is one of the four teams chosen. (c) \(A\) is not one of the four teams chosen.

Suppose that you roll a single die. If an odd number (1,3,0 5) comes up, you win the amount of your roll (\$1, \$3, or \$5 respectively). If an even number \((2,4,\) or 6\()\) comes up, you have to pay the house the amount of your roll \((\$ 2, \$ 4,\) or \(\$ 6\) respectively). (a) Find the expected payoff for this game. (b) Is this a fair game? Explain.

Using set notation, write out the sample space for each of the following random experiments: (a) Three runners \((A, B,\) and \(C\) ) are running in a race (assume that there are no ties). The observation is the order in which the three runners cross the finish line. (b) Four runners \((A, B, C,\) and \(D)\) are running in a qualifying race (assume that there are no ties). The top two finishers qualify for the finals. The observation is the pair of runners that qualify for the finals.

Nine people (four men and five women) line up at a checkout stand in a grocery store. (a) In how many ways can they line up if all five women must be at the front of the line? (b) In how many ways can they line up if they must alternate woman, man, woman, man, and so on?

There are 500 tickets sold in a raffle. If you have three of these tickets and five prizes are to be given, what is the probability that you will win at least one prize? (Give your answer in symbolic notation.
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