Question

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

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

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.

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)

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 \(\frac{4}{10}= 0.4\).

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.