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Chapter 7: Problem 26

A pair of fair dice is rolled. What is the probability that a. The sum of the numbers shown uppermost is less than 5 ? b. At least one 6 is rolled?

Short Answer

Expert verified
a. The probability that the sum of numbers shown uppermost is less than 5 is \( \frac{1}{6} \). b. The probability that at least one 6 is rolled is \( \frac{11}{36} \).

Step by step solution

01

Calculating the probability that the sum of numbers is less than 5

To solve this problem, first, identify all combinations that would result in a sum less than 5. The possible sums are 2, 3 and 4. - There is one scenario with a sum of 2, which is when both dice show 1. - For a sum of 3, there are two scenarios: 1 and 2, 2 and 1. - For a sum of 4, there are three scenarios: 1 and 3, 3 and 1, 2 and 2. Therefore, there are six favorable outcomes. Hence, the probability is \( \frac{6}{36} = \frac{1}{6}\).
02

Calculating the probability of at least one 6 being rolled

All possible outcomes where at least one die shows 6 should be considered. - There are six scenarios where the first die shows 6 and the second die shows 1 to 6. - Similarly, there are six scenarios where the second die shows 6 and the first die shows 1 to 6. - But since both dice showing 6 is included in both the above cases, we should subtract it so that we only count it once to avoid duplication. Therefore, there are 11 favorable outcomes (6 + 6 - 1). Hence, the probability is \( \frac{11}{36} \). So, the answer to the problems are: a. The probability that the sum of numbers shown uppermost is less than 5 is \( \frac{1}{6} \). b. The probability that at least one 6 is rolled is \( \frac{11}{36} \).

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Most popular questions from this chapter

In a survey conducted to see how long Americans keep their cars, 2000 automobile owners were asked how long they plan to keep their present cars. The results of the survey follow: $$ \begin{array}{cc} \hline \text { Years Car Is Kept, } \boldsymbol{x} & \text { Respondents } \\ \hline 0 \leq x<1 & 60 \\ \hline 1 \leq x<3 & 440 \\ \hline 3 \leq x<5 & 360 \\ \hline 5 \leq x<7 & 340 \\ \hline 7 \leq x<10 & 240 \\ \hline 10 \leq x & 560 \\ \hline \end{array} $$ Find the probability distribution associated with these data. What is the probability that an automobile owner selected at random from those surveyed plans to keep his or her present car a. Less than \(5 \mathrm{yr}\) ? b. 3 yr or more?

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or \(7 ?\) Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?

Asurvey involving 400 likely Democratic voters and 300 likely Republican voters asked the question: Do you support or oppose legislation that would require registration of all handguns? The following results were obtained: $$\begin{array}{lcc} \hline \text { Answer } & \text { Democrats, \% } & \text { Republicans, \% } \\\ \hline \text { Support } & 77 & 59 \\ \hline \text { Oppose } & 14 & 31 \\ \hline \text { Don't know/refused } & 9 & 10 \\ \hline \end{array}$$ If a randomly chosen respondent in the survey answered "oppose," what is the probability that he or she is a likely Democratic voter?

Propuct Reuasiumr The proprietor of Cunningham's Hardware Store has decided to install floodlights on the premises as a measure against vandalism and theft. If the probability is \(.01\) that a certain brand of floodlight will burn out within a year, find the minimum number of floodlights that must be installed to ensure that the probability that at least one of them will remain functional for the whole year is at least .99999. (Assume that the floodlights operate independently.)

A halogen desk lamp produced by Luminar was found to be defective. The company has three factories where the lamps are manufactured. The percentage of the total number of halogen desk lamps produced by each factory and the probability that a lamp manufactured by that factory is defective are shown in the accompanying table. What is the probability that the defective lamp was manufactured in factory III? $$ \begin{array}{ccc} \hline & & \text { Probability of } \\ \text { Factory } & \text { Percent of } & \text { Defective } \\ \text { Total Production } & \text { Component } \\ \hline \text { I } & 35 & .015 \\ \hline \text { II } & 35 & .01 \\ \hline \text { III } & 30 & .02 \\ \hline \end{array} $$
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