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Chapter 3: Problem 21

The sample space for tossing three coins is $$ S=\\{h h h, h h t, h t h, h t t, t h h, t h t, t t h, t t t\\} $$ a. List the outcomes that correspond to the statement "All the coins are heads." b. List the outcomes that correspond to the statement "Not all the coins are heads." C. List the outcomes that correspond to the statement "All the coins are not heads."

Short Answer

Expert verified
a. {hhh} b. {hht, hth, htt, thh, tht, tth, ttt} c. {hht, hth, htt, thh, tht, tth, ttt}

Step by step solution

01

Understanding the Sample Space

The sample space given is \( S = \{hhh, hht, hth, htt, thh, tht, tth, ttt\} \). This represents all possible outcomes when tossing three coins. Each outcome is a sequence of three letters where 'h' stands for heads and 't' represents tails.
02

Identify Outcomes for 'All Coins are Heads'

For all coins to be heads, each letter in the sequence must be 'h'. Thus, the corresponding outcome is \( \{hhh\} \).
03

Identify Outcomes for 'Not All Coins are Heads'

'Not all coins are heads' means at least one coin must not be heads. Start from the sample space \( S \) and exclude the outcome \( hhh \). The remaining outcomes are \( \{hht, hth, htt, thh, tht, tth, ttt\} \).
04

Identify Outcomes for 'All Coins are Not Heads'

For 'all coins are not heads', at least one coin must be tails. \( ttt \) cannot be considered as it involves only tails. Thus, the outcomes are the same as in the previous step: \( \{hht, hth, htt, thh, tht, tth, ttt\} \).

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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 concept of a "sample space" is crucial. It represents the set of all possible outcomes of a particular experiment. In our case, when we talk about tossing three coins, each coin has two possible outcomes: heads or tails. This results in a variety of sequences, each combination representing a different possible outcome.Consider this:
  • If coin 1 is heads, coin 2 is heads, and coin 3 is heads, the combined outcome is written as "hhh".
  • If coin 1 is tails, coin 2 is heads, and coin 3 is tails, the outcome is "tht".
Thus, the sample space for tossing three coins is a collection of all these combinations: \[S = \{hhh, hht, hth, htt, thh, tht, tth, ttt\}\] Understanding the full sample space helps in analyzing probabilities since any event's probability is calculated considering these total potential outcomes.
Outcome Analysis
Outcome analysis involves examining the results of a sample space to understand various scenarios and their respective probabilities.For instance, let's delve into the exercise:
  • All coins are heads: Here, we look from the sample space for sequences where each coin shows heads. Hence, the only outcome is \( \{hhh\} \).
  • Not all coins are heads: This outcome means almost the entire sample space except 鈥渉hh鈥�. So, the outcomes are \( \{hht, hth, htt, thh, tht, tth, ttt\} \).
  • All coins are not heads: It is almost similar to the previous scenario, where at least one of the coins must not be a head. The resulting outcomes are also \( \{hht, hth, htt, thh, tht, tth, ttt\} \).
By performing outcome analysis, we can categorize different possible results and calculate their probability, crucial for informed decision-making in probability theory.
Coin Tossing Experiment
The coin tossing experiment is a classic example used in probability theory to foundationally explain random events and outcomes. When you toss a single coin, there are just two possible outcomes:
  • Heads (H)
  • Tails (T)
However, when multiple coins are involved, as in this exercise, the complexity increases. With each additional coin, the number of possible outcomes doubles. Let's see why:
  • **One Coin:** 2 outcomes (H, T)
  • **Two Coins:** 4 outcomes (HH, HT, TH, TT)
  • **Three Coins:** 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT)
This exponential increase in possible outcomes arises because each coin impacts the result independently. Understanding how these outcomes combine in a sample space through experiments like coin tossing enables us to predict probabilities across more complex, random events effortlessly.

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Most popular questions from this chapter

The sample space that describes the two-way classification of citizens according to gender and opinion on a political issue is $$ S=\\{m f, m a, m n, f f, f a, f n\\}, $$ where the first letter denotes gender \((m:\) male, \(f:\) female \()\) and the second opinion \((f:\) for, \(a\) : against, \(n:\) neutral). For each of the following events in the experiment of selecting a citizen at random, state the complement of the event in the simplest possible terms, then find the outcomes that comprise the event and its complement. a. The person is male. b. The person is not in favor. c. The person is either male or in favor. d. The person is female and neutral.

The city council of a particular city is composed of five members of party \(A,\) four members of party \(B,\) and three independents. Two council members are randomly selected to form an investigative committee. a. Find the probability that both are from party \(A\). b. Find the probability that at least one is an independent. c. Find the probability that the two have different party affiliations (that is, not both \(A,\) not both \(B\), and not both independent).

A special deck of 16 cards has 4 that are blue, 4 yellow, 4 green, and 4 red. The four cards of each color are numbered from one to four. A single card is drawn at random. Find the following probabilities. a. The probability that the card drawn is a two or a four. b. The probability that the card is a two or a four, given that it is not a one. c. The probability that the card is a two or a four, given that it is either a two or a three. d. The probability that the card is a two or a four, given that it is red or green.

A manufacturer examines its records over the last year on a component part received from outside suppliers. The breakdown on source (supplier \(A,\) supplier \(B\) ) and quality \((H:\) high \(, U:\) usable, \(D:\) defective \()\) is shown in the two-way contingency table. $$ \begin{array}{|c|c|c|c|} \hline & H & U & D \\ \hline A & 0.6937 & 0.0049 & 0.0014 \\ \hline B & 0.2982 & 0.0009 & 0.0009 \\ \hline \end{array} $$ The record of a part is selected at random. Find the probability of each of the following events. a. The part was defective. b. The part was either of high quality or was at least usable, in two ways: (i) by adding numbers in the table, and (ii) using the answer to (a) and the Probability Rule for Complements. c. The part was defective and came from supplier \(B\). d. The part was defective or came from supplier \(B\), in two ways: by finding the cells in the table that correspond to this event and adding their probabilities, and (ii) using the Additive Rule of Probability.

Compute the following probabilities in connection with the roll of a single fair die. a. The probability that the roll is even. b. The probability that the roll is even, given that it is not a two. c. The probability that the roll is even, given that it is not a one.
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