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Chapter 4: Problem 86

In a statistics class of 42 students, 28 have volunteered for community service in the past. If two students are selected at random from this class, what is the probability that both of them have volunteered for community service in the past? Draw a tree diagram for this problem.

Short Answer

Expert verified
The probability that both randomly picked students are among the volunteers is \( \frac{C(28, 2)}{C(42, 2)} = 0.45 \). So, there is roughly 45% chance that two randomly picked students have both volunteered for community service. The tree diagram offers a visual representation of this setup.

Step by step solution

01

Calculate the total number of ways to select two students

We are given a class of 42 students and we are asked to calculate the probability of selecting two specific students. The number of ways this can be done is given by the combination formula \( C(n, r) = \frac{n!} {r!(n-r)!} \) where 'n' is the total number of items, 'r' is the number of items to choose, and '!' denotes the factorial of a number. In this problem, 'n' would be 42 (the total number of students) and 'r' would be 2 (the number of students being selected). We obtain \( C(42, 2) = \frac{42!}{2!(42-2)!} \)
02

Calculate the desired number of outcomes

We want both students to be among those who have volunteered. We are told that there are 28 students who have volunteered. So we want to calculate the number of ways to choose 2 students from these 28 volunteers. This is again calculated using the combination formula. This time 'n' would be 28 and 'r' would be 2. We obtain \( C(28, 2) = \frac{28!}{2!(28-2)!} \)
03

Calculate the probability

The probability is equal to the desirable outcomes divided by the total outcomes. We divide the result from step 2 by the result from step 1. This will give us the probability.
04

Draw a tree diagram

Start with one node symbolizing the total number of students. Draw two branches, one representing the volunteers and the other the students who haven't volunteered. From both of these nodes, draw two more nodes each, again representing volunteers and non-volunteers respectively. This gives you a visual representation of all possible scenarios when picking two students randomly.

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Most popular questions from this chapter

The following table gives a two-way classification of all basketball players at a state university who began their college careers between 2001 and 2005, based on gender and whether or not they graduated \(\begin{array}{lcc} \hline & \text { Graduated } & \text { Did Not Graduate } \\ \hline \text { Male } & 126 & 55 \\ \text { Female } & 133 & 32 \\ \hline \end{array}\) a. If one of these players is selected at random, find the following probabilities. i. \(P(\) female and graduated \()\) ii. \(P(\) male and did not graduate \()\) b. Find \(P\) (graduated and did not graduate). Is this probability zero? If yes, why?

In a sample survey, 1800 senior citizens were asked whether or not they have ever been victimized by a dishonest telemarketer. The following table gives the responses by age group. $$\begin{array}{l|llcc} & & & \begin{array}{c} \text { Have Been } \\ \text { Victimized } \end{array} & \begin{array}{c} \text { Have Never } \\ \text { Been Victimized } \end{array} \\ \hline & 60-69 & \text { (A) } & 106 & 698 \\ \text { Age } & 70-79 & \text { (B) } & 145 & 447 \\ & 80 \text { or over (C) } & 61 & 343 \\ \hline \end{array}$$ Suppose one person is randomly selected from these senior citizens. Find the following probabilities a. \(P(\) have been victimized or \(\mathrm{B}\) ) b. \(P(\) have never been victimized or \(\mathrm{C}\) )

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of majon iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.

Suppose that \(20 \%\) of all adults in a small town live alone, and \(8 \%\) of the adults live alone and have at least one pet. What is the probability that a randomly selected adult from this town has at least one pet given that this adult lives alone?

Briefly explain the three approaches to probability. Give one example of each approach.
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