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Chapter 4: Problem 26

A die is rolled once. What is the probability that a. a number less than 5 is obtained? b. a number 3 to 6 is obtained?

Short Answer

Expert verified
The probability that a number less than 5 is obtained when a dice is rolled once is \(\frac{2}{3}\). The probability that a number from 3 to 6 is obtained when a dice is rolled once is also \(\frac{2}{3}\).

Step by step solution

01

Determine the total number of outcomes

When a fair dice is rolled once, it has six possible outcomes which are 1, 2, 3, 4, 5, and 6. So, the total number of outcomes is 6.
02

Calculate probability for condition a

The condition a asks for the probability that a number less than 5 is obtained. Numbers less than 5 are 1, 2, 3, and 4. So, there are 4 outcomes that satisfy the condition a. Therefore, the probability for condition a can be calculated as the number of desirable outcomes divided by the total number of outcomes which is \(\frac{4}{6}= \frac{2}{3}\).
03

Calculate probability for condition b

The condition b asks for the probability that a number 3 to 6 is obtained. Numbers between 3 and 6 inclusive are 3, 4, 5, and 6. So, there are 4 outcomes that satisfy the condition b. Therefore, the probability for condition b can be calculated similarly as the number of desirable outcomes divided by the total number of outcomes which is \(\frac{4}{6}= \frac{2}{3}\).

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Most popular questions from this chapter

Consider the following games with two dice. a. A gambler is going to roll a die four times. If he rolls at least one 6, you must pay him \(\$ 5 .\) If he fails to roll a 6 in four tries, he will pay you \(\$ 5\). Find the probability that you must pay the gambler. Assume that there is no cheating. b. The same gambler offers to let you roll a pair of dice 24 times. If you roll at least one double 6 , he will pay you \(\$ 10\). If you fail to roll a double 6 in 24 tries, you will pay him \(\$ 10\). The gambler says that you have a better chance of winning because your probability of success on each of the 24 rolls is \(1 / 36\) and you have 24 chances. Thus, he says, your probability of winning \(\$ 10\) is \(24(1 / 36)=2 / 3\). Do you agree with this analysis? If so, indicate why. If not, point out the fallacy in his argument, and then find the correct probability that you will win.

Five hundred employees were selected from a city's large private companies and asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$\begin{array}{llc} \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array}$$ a. Suppose one employee is selected at random from these 500 employees. Find the following probabilities. i. Probability of the intersection of events "woman" and "yes" ii. Probability of the intersection of events "no" and "man" b. Mention what other joint probabilities you can calculate for this table and then find them. You may draw a tree diagram to find these probabilities.

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A production system has two production lines; each production line performs a two-part process, and each process is completed by a different machine. Thus, there are four machines, which we can identify as two first-level machines and two second-level machines. Each of the first-level machines works properly \(98 \%\) of the time, and each of the second-level machines works properly \(96 \%\) of the time. All four machines are independent in regard to working properly or breaking down. Two products enter this production system, one in each production line a. Find the probability that both products successfully complete the two-part process (i.e., all four machines are working properly). b. Find the probability that neither product successfully completes the two- part process (i.e., at least one of the machines in each production line is not working properly).
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