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Chapter 6: Problem 102

A box contains five slips of paper, marked \(\$ 1, \$ 1, \$ 1, \$ 10,\) and \(\$ 25 .\) The winner of a contest selects two slips of paper at random and then gets the larger of the dollar amounts on the two slips. Define a random variable \(w\) by \(w=\) amount awarded. Determine the probability distribution of \(w\). (Hint: Think of the slips as numbered \(1,2,3,4,\) and \(5 .\) An outcome of the experiment will consist of two of these numbers.)

Short Answer

Expert verified
The probability distribution of \(w\) is:\n\(w=1\) has a probability of 0.3\n\(w=10\) has a probability of 0.3\n\(w=25\) has a probability of 0.4

Step by step solution

01

Setting Up

Observe that the box has five slips: three slips marked with \$1, one slip marked with \$10, and one slip marked with \$25.\nThere are two stages in which one slip is drawn in each stage. Thus, an outcome of the experiment will consist of drawing two slips.
02

Identify Possible Outcomes

Outcomes are divided into three categories based on the dollar amount awarded: \n1) \$1: This happens when two \$1 slips are drawn out. (Combinations are: {1,2}, {1,3}, {2,3} - Total of 3 combinations)\n2) \$10: This happens when one \$10 slip and one \$1 slip are drawn. (Combinations are: {4,1}, {4,2}, {4,3} - Total of 3 combinations)\n3) \$25: This happens when either two \$25 slips are drawn or one \$25 slip and another slip (either \$10 or \$1). (Combinations are: {5,4}, {5,1}, {5,2}, {5,3} - Total of 4 combinations)
03

Calculate Probabilities

Now, calculate the probability for each outcome:\n1) The probability that \$1 is awarded is 3 out of a total of 10 possible combinations, which is \(\frac{3}{10}= 0.3\).\n2) The probability that \$10 is awarded is 3 out of a total of 10 possible combinations, which is \(\frac{3}{10}= 0.3\).\n3) The probability that \$25 is awarded is 4 out of a total of 10 possible combinations, which is \(\frac{4}{10}= 0.4\).
04

Establish the Probability Distribution

The final step of the problem involves ensuring that the sum of the probabilities is equal to 1 and summarizing the results into a table with the possible values of the random variable \(w\) and their associated probabilities.

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Most popular questions from this chapter

Consider the variable \(x=\) time required for a college student to complete a standardized exam. Suppose that for the population of students at a particular university, the distribution of \(x\) is well approximated by a normal curve with mean 45 minutes and standard deviation 5 minutes. a. If 50 minutes is allowed for the exam, what proportion of students at this university would be unable to finish in the allotted time? b. How much time should be allowed for the exam if you wanted \(90 \%\) of the students taking the test to be able to finish in the allotted time? c. How much time is required for the fastest \(25 \%\) of all tudents to complete the exam

A restaurant has four bottles of a certain wine in stock. The wine steward does not know that two of these bottles (Bottles 1 and 2) are bad. Suppose that two bottles are ordered, and the wine steward selects two of the four bottles at random. Consider the random variable \(x=\) the number of good bottles among these two. a. One possible experimental outcome is (1,2) (Bottles 1 and 2 are selected) and another is (2,4) . List all possible outcomes. b. What is the probability of each outcome in Part (a)? c. The value of \(x\) for the (1,2) outcome is 0 (neither selected bottle is good), and \(x=1\) for the outcome (2,4) . Determine the \(x\) value for each possible outcome. Then use the probabilities in Part (b) to determine the probability distribution of \(x\). (Hint: See Example 6.5 )

Let \(x\) denote the IQ of an individual selected at random from a certain population. The value of \(x\) must be a whole number. Suppose that the distribution of \(x\) can be approximated by a normal distribution with mean value 100 and standard deviation 15. Approximate the following probabilities. (Hint: See Example 6.32 ) a. \(P(x=100)\) b. \(P(x \leq 110)\) c. \(P(x<110)\) (Hint: \(x<110\) is the same as \(x \leq 109\).) d. \(P(75 \leq x \leq 125)\)

An appliance dealer sells three different models of freezers having \(13.5,15.9,\) and 19.1 cubic feet of storage space. Let \(x=\) the amount of storage space purchased by the next customer to buy a freezer. Suppose that \(x\) has the following probability distribution: $$ \begin{array}{lrrr} x & 13.5 & 15.9 & 19.1 \\ p(x) & 0.2 & 0.5 & 0.3 \end{array} $$ a. Calculate the mean and standard deviation of \(x\). (Hint: See Example 6.15\()\) b. Give an interpretation of the mean and standard deviation of \(x\) in the context of observing the outcomes of many purchases.

Seventy percent of the bicycles sold by a certain store are mountain bikes. Among 100 randomly selected bike purchases, what is the approximate probability that a. At most 75 are mountain bikes? (Hint: See Example 6.33 ) b. Between 60 and 75 (inclusive) are mountain bikes? c. More than 80 are mountain bikes? d. At most 30 are not mountain bikes?
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