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Chapter 7: Problem 43

Determine whether the given experiment has a sample space with equally likely outcomes. Two fair dice are rolled, and the sum of the numbers appearing uppermost is recorded.

Short Answer

Expert verified
In this experiment, where two fair dice are rolled and the sum of the numbers appearing uppermost is recorded, the sample space does not have equally likely outcomes. This is because not all sums occur with the same frequency in the sample space.

Step by step solution

01

Identify Possible Outcomes

Since there are two dice being rolled and each die has six possible outcomes (1-6), we can create a sample space with six rows and six columns, where each row and each column represents the outcomes of one die. The cells represent the sum of the numbers rolled.
02

Calculate the Sum of Uppermost Faces

Now, for each cell in this sample space, calculate the sum of the numbers rolled for both dice. To do this, add the row number to the column number for each cell. For example: - Row 1, Column 1: \(1 + 1 = 2\), - Row 1, Column 2: \(1 + 2 = 3\), - Row 1, Column 3: \(1 + 3 = 4\) and so on. Continue this process for all cells.
03

Count the Number of Occurrences for Each Sum

After calculating the sum in each cell, count the number of times each sum appears in the sample space. This will help determine whether each sum is equally likely to happen. For example, the sum: - 2 occurs once, - 3 occurs twice, - 4 occurs three times, - 5 occurs four times, - 6 occurs five times, - 7 occurs six times, - 8 occurs five times, - 9 occurs four times, - 10 occurs three times, - 11 occurs twice, - 12 occurs once.
04

Analyze the Probability of Each Sum

Based on the counts of the sums in the sample space, we can see that not all sums occur with the same frequency. Therefore, the outcomes of this experiment (the sums) are not equally likely. #Conclusion# In this experiment, where two fair dice are rolled and the sum of the numbers appearing uppermost is recorded, the sample space does not have equally likely outcomes.

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Most popular questions from this chapter

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}\right\\}\) be the sample space associated with an experiment having the following probability distribution: $$\begin{array}{lcccccc} \hline \text { Outcome } & s_{1} & s_{2} & s_{3} & s_{4} & s_{5} & s_{6} \\ \hline \text { Probability } & \frac{1}{12} & \frac{1}{4} & \frac{1}{12} & \frac{1}{6} & \frac{1}{3} & \frac{1}{12} \\ \hline \end{array}$$ Find the probability of the event: a. \(A=\left\\{s_{1}, s_{3}\right\\}\) b. \(B=\left\\{s_{2}, s_{4}, s_{5}, s_{6}\right\\}\) c. \(C=S\)

A time study was conducted by the production manager of Universal Instruments to determine how much time it took an assembly worker to complete a certain task during the assembly of its Galaxy home computers. Results of the study indicated that \(20 \%\) of the workers were able to complete the task in less than \(3 \mathrm{~min}\), \(60 \%\) of the workers were able to complete the task in \(4 \mathrm{~min}\) or less, and \(10 \%\) of the workers required more than \(5 \mathrm{~min}\) to complete the task. If an assembly-line worker is selected at random from this group, what is the probability that a. He or she will be able to complete the task in 5 min or less? b. He or she will not be able to complete the task within 4 min? c. The time taken for the worker to complete the task will be between 3 and 4 min (inclusive)?

In a past presidential election, it was estimated that the probability that the Republican candidate would be elected was \(\frac{3}{5}\), and therefore the probability that the Democratic candidate would be elected was \(\frac{2}{5}\) (the two Independent candidates were given no chance of being elected). It was also estimated that if the Republican candidate were elected, the probability that a conservative, moderate, or liberal judge would be appointed to the Supreme Court (one retirement was expected during the presidential term) was \(\frac{1}{2}, \frac{1}{3}\), and \(\frac{1}{6}\), respectively. If the Democratic candidate were elected, the probabilities that a conservative, moderate, or liberal judge would be appointed to the Supreme Court would be \(\frac{1}{8}, \frac{3}{8}\), and \(\frac{1}{2}\), respectively. A conservative judge was appointed to the Supreme Court during the presidential term. What is the probability that the Democratic candidate was elected?

In "The Numbers Game," a state lottery, four numbers are drawn with replacement from an urn containing balls numbered \(0-9\), inclusive. Find the probability that a ticket holder has the indicated winning ticket. Two specified, consecutive digits in exact order (the first two digits, the middle two digits, or the last two digits)

Determine whether the given experiment has a sample space with equally likely outcomes. A loaded die is rolled, and the number appearing uppermost on the die is recorded.
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