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Chapter 4: Problem 81

In a political science class of 35 students, 21 favor abolishing the electoral college and thus electing the President of the United States by popular vote. If two students are selected at random from this class, what is the probability that both of them favor abolition of the electoral college? Draw a tree diagram for this problem.

Short Answer

Expert verified
The probability that both students, selected at random, favor abolition of the electoral college is \(\frac{6}{17}\).

Step by step solution

01

Understanding the problem

The class has 35 students, out of which 21 favor abolition of the electoral college. If two students are selected randomly from the class, we need to find out the probability that both of these students favor abolition.
02

Calculating the probability for the first student

To find out the probability that the first student favors abolition, divide the number of students who favor abolition by the total number of students in the class. Hence, the probability is \(\frac{21}{35}\) or \(\frac{3}{5}\).
03

Calculating the probability for the second student

If the first selected student is in favor of abolition, the class now consists of 34 students of which 20 favor abolition. Hence, the probability the second student favors abolition, knowing that one student in favor of abolition has already been selected, is \(\frac{20}{34}\) or \(\frac{10}{17}\).
04

Calculating the overall probability

The overall probability of both students favoring abolition is the product of the individual probabilities. Hence, the overall probability is \(\frac{3}{5} * \frac{10}{17}\) or \(\frac{30}{85}\). After we reduce the fraction, we get \(\frac{6}{17}\) as the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tree Diagram
A tree diagram is a visual tool used in probability and statistics to map out all possible outcomes of an event in a structured and systematic way. Imagine it as a branching diagram that resembles a family tree, where each branch represents a possible outcome. For the political science class exercise, each branch shows the sequence of students selected and whether they favor abolition of the electoral college.

In our problem:
  • Start with two main branches: one for picking a student in favor (21 out of 35) and one not in favor (14 out of 35).
  • From each of these branches, more branches extend representing the selection of the second student with adjusted probabilities.
  • The final branches show the outcomes: both students in favor, one in favor, and none in favor.
These diagrams not only simplify the process of finding outcomes but also help to clearly visualize the sequential steps in calculating probabilities.
Sample Space
The sample space in probability refers to the set of all possible outcomes of a particular experiment. In simpler terms, it's like the "menu" of outcomes you can pick from for a given scenario.

For the class exercise:
  • The sample space includes every combination of two students 鈥� whether both are in favor, one is, or none is in favor.
  • Each of these combinations is a part of the overall event that may occur when selecting two students.
  • In terms of probabilities, knowing our sample space helps in calculating specific probabilities, like the likelihood of picking both students who favor abolition.
Understanding the sample space is crucial because it sets a foundation on which probability calculations rest.
Probability of Independent Events
Independent events are those where the occurrence of one event doesn't affect the occurrence of another. But in the given problem, we're dealing with dependent events since the selection of one student influences the pool from which the second student is chosen.

However, it is good to note that if the events were independent:
  • The probability of both events happening is simply the product of their individual probabilities.
  • For instance, if the selection of the first student didn't alter the options, you'd multiply the probability of both separately.
By contrast, in our example, you must adjust probabilities with each selection, as each choice affects subsequent ones.
Conditional Probability
Conditional probability refers to the probability of an event occurring given that another event has already occurred. It's about refining the probability after gaining some new information. In the given class exercise, this concept is essential for understanding why probabilities change after the choice of the first student.

For example:
  • The probability that the second student chosen favors abolition depends on the first student's selection.
  • The revised probability for the second student becomes a conditional probability, hence being \( \frac{20}{34} \) once the first student is chosen having favored abolition.
Utilizing conditional probability ensures that we take into account how preceding events impact the outcomes in sequential selections.

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Most popular questions from this chapter

Consider the following addition rule to find the probability of the union of two events \(A\) and \(B\) : $$ P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B) $$ When and why is the term \(P(A\) and \(B\) ) subtracted from the sum of \(P(A)\) and \(P(B)\) ? Give one example where you might use this formula.

How many different outcomes are possible for four rolls of a die?

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 59, and then picks a Powerball number from 1 through \(39 .\) The lottery organization randomly draws 5 different white balls from 59 balls numbered 1 through 59 , and then randomly picks a Powerball number from 1 through \(39 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.) b. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about \(\$ 200,000\). Find the probability that a player who buys one ticket will win this prize.

Refer to Exercise 4.48. A 2010-2011 poll conducted by Gallup (www.gallup.com/poll/148994/ Emotional-Health-Higher-Among-Older- Americans.aspx) examined the emotional health of a large number of Americans. Among other things, Gallup reported on whether people had Emotional Health Index scores of 90 or higher, which would classify them as being emotionally well-off. The report was based on a survey of 65,528 people in the age group \(35-44\) years and 91,802 people in the age group \(65-74\) years. The following table gives the results of the survey, converting percentages to frequencies. $$ \begin{array}{lcc} \hline & \text { Emotionally Well-Off } & \text { Emotionally Not Well-Off } \\\ \hline \text { 35-44 Age group } & 16,016 & 49,512 \\ \text { 65-74 Age group } & 32,583 & 59,219 \\ \hline \end{array} $$ a. Suppose that one person is selected at random from this sample of 157,330 Americans. Find the following probabilities. i. \(P(35-44\) age group and emotionally not well-off \()\) ii. \(P(\) emotionally well-off and \(65-74\) age group \()\) b. Find the joint probability of the events \(35-44\) age group and \(65-74\) age group. Is this probability zero? Explain why or why not.

Given that \(A\) and \(B\) are two independent events, find their joint probability for the following. a. \(P(A)=.29 \quad\) and \(\quad P(B)=.65\) b. \(P(A)=.03\) and \(P(B)=.28\)
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