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Chapter 5: Problem 28

Which of the following are binomial experiments? Explain why. a. Rolling a die 10 times and observing the number of spots b. Rolling a die 12 times and observing whether the number obtained is even or odd c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when \(54 \%\) of all voters are known to be in favor of this proposition.

Short Answer

Expert verified
All the presented scenarios (a, b, and c) are binomial experiments because they meet the criteria set for binomial experiments. These are: each trial is an independent event, the probability of success is the same for each trial, the number of trials is fixed, and the outcome of each trial can only be either 'success' or 'failure'.

Step by step solution

01

Analyze Scenario a

In scenario a, the process of rolling a die and observing the numbers of spots can be considered a binomial experiment because: 1. Each roll of the die is an independent event. 2. For each roll, the probability of each outcome (1 to 6) is equal (\(\frac{1}{6}\)). 3. The number of trials (rolling the die) is fixed at 10. 4. The potential outcomes are a 'success' (the dice lands on the spot you are observing) or 'failure' (the dice lands on a spot different than the one you are observing). So, scenario a is a binomial experiment.
02

Analyze Scenario b

In scenario b, the process of rolling a die and observing whether the number obtained is even or odd can also be considered a binomial experiment because: 1. Each roll of the die is an independent event. 2. The probability of getting an even or an odd number is equal (\(\frac{1}{2}\)) in each roll. 3. The number of trials (rolling the die) is fixed at 12. 4. The potential outcomes are a 'success' (the dice lands on an even or odd number, depending on what is considered success) or 'failure' (the dice lands on the opposite of what was considered success). So, scenario b is also a binomial experiment.
03

Analyze Scenario c

In scenario c, the process of selecting voters from a population and observing whether or not they favor a certain proposition can also be considered a binomial experiment because: 1. Each selection of a voter is an independent event. 2. The probability of success (54% favoring the proposition) is the same for each selection. 3. The number of trials (voters selected) can be considered a fixed number. 4. The potential outcomes are a 'success' (a voter favors the proposition) or 'failure' (a voter does not favor the proposition). So, scenario c is also a binomial experiment.

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Key Concepts

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

Independent Events
In a binomial experiment, independent events are crucial. This means that the outcome of one trial does not affect the outcome of another. For example, when you roll a die, one roll is separate from the next. Similarly, if you're selecting voters to see if they favor a proposition, each voter's choice is considered independently from others. This independence ensures that the experiment's outcome is fair and unbiased.
Understanding independence helps eliminate the chance of patterns influencing results, which is fundamental to analyzing these types of experiments accurately.
Probability
Probability is a measure of how likely an event is to occur. In binomial experiments, we often deal with straightforward scenarios where each trial has two possible outcomes.
For example, when rolling a die, if you are considering either getting an even number or not, the probability of getting an even number is \( \frac{1}{2} \). If selecting voters, and 54% favor a proposition, then the probability for each selection is 0.54.
  • This constant probability is a hallmark of binomial experiments.
  • It helps in predicting how often we might expect a particular outcome.
Analyzing probability is essential to understanding the expected results in any binomial setup.
Fixed Number of Trials
A fixed number of trials means you know beforehand how many times the experiment will be conducted. This is a defining feature of binomial experiments.
Whether you're rolling a die 10 times or selecting a certain number of voters, the number of trials remains the same throughout.
  • It provides a framework for calculating expected outcomes.
  • Without a set number of trials, analyzing results would be unpredictable.
Thus, having a fixed number of trials simplifies the process of calculating probabilities and assessing results.
Success and Failure Outcomes
In binomial experiments, each trial results in one of two outcomes: success or failure. These outcomes are specific and predefined depending on the context.
For instance, if we define success as rolling an even number on a die, a failure would be rolling an odd number. Similarly, in voter selection, success could mean the voter favors a proposition, while failure means they do not.
  • This binary outcome setup is central to binomial experiments.
  • It helps in applying the binomial formula to determine probabilities.
Defining these outcomes clearly is crucial for analyzing results correctly.

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Most popular questions from this chapter

Explain the hypergeometric probability distribution. Under what conditions is this probability distribution applied to find the probability of a discrete random variable \(x\) ? Give one example of an application of the hypergeometric probability distribution.

The following table gives the probability distribution of a discrete random variable \(x\). $$ \begin{array}{l|ccccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .11 & .19 & .28 & .15 & .12 & .09 & .06 \\ \hline \end{array} $$ Find the following probabilities. a. \(P(3)\) b. \(P(x \leq 2)\) c. \(P(x \geq 4)\) d. \(P(1 \leq x \leq 4)\) e. Probability that \(x\) assumes a value less than 4 f. Probability that \(x\) assumes a value greater than 2 g. Probability that \(x\) assumes a value in the interval 2 to 5

Let \(x\) be the number of houses sold per month by a real estate agent. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|cccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & .08 & .12 & .32 & .28 & .12 & .08 \\ \hline \end{array} $$ Calculate the mean and standard deviation of this probability distribution and give a brief interpretation of the value of the mean.

In a group of 12 persons, 3 are left-handed. Suppose that 2 persons are randomly selected from this group. Let \(x\) denote the number of left-handed persons in this sample. Write the probability distribution of \(x\). You may draw a tree diagram and use it to write the probability distribution. (Hint: Note that the selections are made without replacement from a small population. Hence, the probabilities of outcomes do not remain constant for each selection.)

Many of you probably played the game "Rock, Paper, Scissors" as a child. Consider the following variation of that game. Instead of two players, suppose three players play this game, and let us call these players \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each player selects one of these three items- Rock, Paper, or Scissors-independent of each other. Player A will win the game if all three players select the same item, for example, rock. Player B will win the game if exactly two of the three players select the same item and the third player selects a different item. Player \(\mathrm{C}\) will win the game if every player selects a different item. If Player B wins the game, he or she will be paid \(\$ 1\). If Player \(\mathrm{C}\) wins the game, he or she will be paid \(\$ 3\). Assuming that the expected winnings should be the same for each player to make this a fair game, how much should Player A be paid if he or she wins the game?
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