Question

Census reports for a city indicate that \(62 \%\) of residents classify themselves as Christian, \(12 \%\) as Jewish, and \(16 \%\) as members of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as nonreligious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach a) all Christians? b) no Jews? c) at least one person who is nonreligious?

Short answer

a) 0.1477; b) 0.5997; c) 0.3439.

Step-by-step solution

Define Probabilities

To solve the problem, first identify the probability of each subset of people. The probability that a person is Christian is \(0.62\). The probability a person is Jewish is \(0.12\). The probability a person is of another religion is \(0.16\). Therefore, the probability of being nonreligious is the total remaining, \(1 - (0.62 + 0.12 + 0.16) = 0.10\).

Probability of All Christians

For part (a), calculate the probability that all four people reached are Christians. The probability of each person being Christian is \(0.62\), and we want this event for four people. Thus, the probability is \(0.62^4\). Calculate \(0.62^4 = 0.1477\).

Probability of No Jews

For part (b), find the probability that none of the four people are Jewish. The probability that a person is not Jewish is \(1 - 0.12 = 0.88\). The probability that none of the four reached are Jewish is \(0.88^4\). Calculate \(0.88^4 = 0.5997\).

Probability of At Least One Nonreligious

For part (c), calculate the probability of reaching at least one person who is nonreligious. First, calculate the probability that none of them are nonreligious, which means all are religious. This is \((0.9)^4 = 0.6561\). The probability of at least one nonreligious person is therefore \(1 - 0.6561 = 0.3439\).

Key concepts

Probability Calculation

Probability is all about measuring chance or likelihood. When dealing with real-life situations, like in our exercise, we often calculate the probability of multiple events happening in sequence.
In this example, we want to know the probability of certain religious demographics among the first four people reached by phone by a polling organization.
A key aspect of probability calculations is understanding independent and dependent events. Here, each person's religion is assumed independent of the others.
This means the probability of reaching a Christian lawyer, for instance, does not affect the probability of the next person’s religion. Let's delve into how these probabilities are determined:

• "Probability of All Christians" required calculating the probability that each of four individuals were Christians. Mathematically, this is expressed as raising the probability of one Christian to the power of four: \( (0.62)^4 \).
• "Probability of No Jews" called for considering that none of the four were Jewish, which involved a similar calculation: \( (0.88)^4 \).
• For "At Least One Nonreligious", the approach is to calculate the complementary event, that is, none being nonreligious (all being religious): \( (0.9)^4 \). Then, subtract from 1 to find the target probability.

Probability helps us understand the chances of reaching people with specific beliefs, providing valuable information for decision making.

Statistics

Statistics is the study of collecting and interpreting data. It allows us to make informed predictions and solve real-world problems, like finding the probability of different religions in a city.
With a large sample, like in a city's population, statistical probabilities enable the estimation of the presence or absence of certain features in individuals. These features, in terms of our exercise, are different religious beliefs.
The likelihood of random calls reaching specific religious groups was calculated using percentages from the census data. This is part of inferential statistics where we draw conclusions from data samples.

• In this exercise, the statistics show that 62% are Christians, 12% Jewish, 16% other religions, and 10% nonreligious.
• The core principle is understanding and applying probabilities to model possible outcomes.

This way, statistics acts as a foundational tool, enabling the prediction of random sampling results, which is crucial in demographic studies.

Religion Demographics

Demographics play an essential role in understanding the makeup of a population. It allows organizations to tailor their approaches and ensures representation across various social factors. Such information is incredibly relevant for institutions conducting surveys, like in our exercise.
Studying religion demographics particularly helps in grasping the diversity within a community. It includes the proportion of individuals adhering to different religions, such as Christianity, Judaism, or others, as well as those who identify as nonreligious.

• In our scenario, 62% are Christians, showcasing a significant part of the population.
• Meanwhile, 12% of Jewish individuals and another 16% belonging to other various religions present diversity.
• The 10% nonreligious further add to understanding societal beliefs.

Understanding these percentages can aid in designing surveys and interpreting results accurately. It ensures inclusivity when collecting insights about public opinions or fulfilling organizational objectives. Demographics thus give us essential insights into who makes up the society, promoting informed decisions and policies.