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Chapter 7: Problem 27

What is the probability of arriving at a traffic light when it is red if the red signal is lit for \(30 \mathrm{sec}\), the yellow signal for \(5 \mathrm{sec}\), and the green signal for \(45 \mathrm{sec}\) ?

Short Answer

Expert verified
The probability of arriving at a traffic light when it is red is \(\frac{3}{8}\).

Step by step solution

01

Find total duration of one complete cycle

To find the total duration of one cycle, we will sum up the duration of all three signals: red, yellow, and green. Let T be the total duration of the cycle: \[T = 30 sec + 5 sec + 45 sec\]
02

Calculate the probability

Now that we have the total duration of one cycle (T), we can find the probability of arriving at the traffic light when it is red. We know that the red signal is lit for 30 seconds, so we can find the probability by dividing the duration of the red signal by the total duration, T: \[\text{Probability} = \frac{\text{Duration of red signal}}{\text{Total duration}}\]
03

Plug in values and compute

Now we can plug in the values for the duration of the red light and the total duration of the cycle: \[\text{Probability} = \frac{30 sec}{30 sec + 5 sec + 45 sec}\]
04

Simplify the expression

Next, we will simplify the expression by adding the durations: \[\text{Probability} = \frac{30 sec}{80 sec}\]
05

Calculate the final probability

Now, we will simplify the fraction to obtain our final probability: \[\text{Probability} = \frac{3}{8}\] Therefore, the probability of arriving at a traffic light when it is red is \(\frac{3}{8}\).

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Most popular questions from this chapter

A nationwide survey conducted by the National Cancer Society revealed the following information. Of 10,000 people surveyed, 3200 were "heavy coffee drinkers" and 160 had cancer of the pancreas. Of those who had cancer of the pancreas, 132 were heavy coffee drinkers. Using the data in this survey, determine whether the events "being a heavy coffee drinker" and "having cancer of the pancreas" are independent events.

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? $$ \begin{array}{ccc} \hline & & \text { Probability of } \\ \text { Factory } & \text { Percent of } & \text { Defective } \\ \text { Total Production } & \text { Component } \\ \hline \text { I } & 35 & .015 \\ \hline \text { II } & 35 & .01 \\ \hline \text { III } & 30 & .02 \\ \hline \end{array} $$

According to a study conducted in 2003 concerning the participation, by age, of \(401(\mathrm{k})\) investors, the following data were obtained: $$ \begin{array}{lccccc} \hline \text { Age } & 20 \mathrm{~s} & 30 \mathrm{~s} & 40 \mathrm{~s} & 50 \mathrm{~s} & 60 \mathrm{~s} \\ \hline \text { Percent } & 11 & 28 & 32 & 22 & 7 \\ \hline \end{array} $$ a. What is the probability that a \(401(\mathrm{k})\) investor selected at random is in his or her 20 s or 60 s? b. What is the probability that a \(401(\mathrm{k})\) investor selected at random is under the age of 50 ?

In a survey on consumer-spending methods conducted in 2006, the following results were obtained: $$\begin{array}{lccccc} \hline & & & & {\text { Debit/ATM }} & \\ \text { Payment Method } & \text { Checks } & \text { Cash } & \text { Credit cards } & \text { cards } & \text { Other } \\ \hline \text { Transactions, \% } & 37 & 14 & 25 & 15 & 9 \\ \hline \end{array}$$ If a transaction tracked in this survey is selected at random, what is the probability that the transaction was paid for a. With a credit card or with a debit/ATM card? b. With cash or some method other than with a check, a credit card, or a debit/ATM card?

A study of the faculty at U.S. medical schools in 2006 revealed that \(32 \%\) of the faculty were women and \(68 \%\) were men. Of the female faculty, \(31 \%\) were full/ associate professors, \(47 \%\) were assistant professors, and \(22 \%\) were instructors. Of the male faculty, \(51 \%\) were full/associate professors, \(37 \%\) were assistant professors, and \(12 \%\) were instructors. If a faculty member at a U.S. medical school selected at random holds the rank of full/associate professor, what is the probability that she is female?
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