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Chapter 4: Problem 65

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?

Short Answer

Expert verified
The probability that a randomly selected adult from the town has at least one pet given that this adult lives alone is 40%.

Step by step solution

01

Understand The Problem

Here the problem is actually asking for the conditional probability. The event B in this case is that an adult lives alone, and event A is that an adult has at least one pet. We want to find P(A|B), or the probability of A given B. In other words, given that an adult lives alone, what is the probability that they have a pet?
02

Formulate Conditional Probability Formula

The formula for conditional probability is P(A|B) = P(A 鈭� B) / P(B). In our problem, event A is that an adult has at least one pet, and event B is that an adult lives alone. So, we need to find P(A 鈭� B) and P(B).
03

Substitute The Known Values

From the problem, we know that the value of P(A 鈭� B) is 0.08 or 8%, which is the probability that an adult lives alone and has a pet. P(B) is given as 0.20 or 20%, which is the probability that an adult lives alone. Plug these values into the formula to get P(A|B) = 0.08 / 0.20.
04

Calculate The Conditional Probability

On calculating P(A|B) = 0.08 / 0.20, we find that P(A|B) equals to 0.40 or 40%.

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics concerned with the analysis of random phenomena. In simple terms, it allows us to predict the likelihood of events happening. These events are often uncertain, and probability theory provides a framework to understand these uncertainties.
  • Mathematical formulas in probability theory help quantify the likelihood of different possible outcomes.
  • This theory is used extensively in fields such as statistics, finance, science, and engineering, among others.
Imagine you're flipping a coin. There are two possible outcomes: heads or tails. Probability helps us predict that each outcome has a fifty percent chance of occurring. In the context of our exercise, probability theory underpins the calculation involving adults living alone and owning pets. These concepts help us simplify complex relationships into manageable and interpretable numbers.
Event Probability
Event probability is the measure of the likelihood that a specific event will occur. An event is any outcome or combination of outcomes from a random experiment. In everyday terms, whenever you talk about the chances of something happening, you're discussing event probability.
  • For example, the event "it will rain tomorrow" might have a probability of 70%. This means there is a fairly high chance of rain.
  • Event probability is denoted by P(E), where E stands for some event.
In the given problem:
  • The probability of event A, an adult having at least one pet, given that they live alone, is what we want to calculate.
  • The prior probability of event B, that an adult lives alone, is given as 20% or 0.20.
Understanding event probability is crucial as it helps one to evaluate different outcomes and make informed decisions in uncertain scenarios.
Conditional Events
Conditional events refer to instances where an event's probability is influenced by the occurrence of another event. This type of conditional probability is especially valuable when trying to determine the likelihood of correlated or dependent events. When we speak of conditional probability, it might sound complex, but it breaks down into simpler steps:
  • Determine the event for which you want to find the probability (e.g., an adult has at least one pet).
  • Identify the condition affecting this probability (e.g., an adult lives alone).
Using the formula for conditional probability, we calculate: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]Where:
  • \(P(A|B)\) is the probability that event A occurs given event B.
  • \(P(A \cap B)\) is the probability that both events A and B occur simultaneously, given as 0.08 or 8%.
  • \(P(B)\) is the probability that event B occurs, given as 0.20 or 20%.
In our exercise, we calculate the probability that an adult, who lives alone, also has a pet, resulting in a conditional probability of 40%.

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Most popular questions from this chapter

In a survey of adults, \(40 \%\) hold the opinion that there will be another housing bubble in the next four to six years. Three adults are selected at random. a. What is the probability that all three adults hold the opinion that there will be another housing bubble in the next four to six years? b. What is the probability that none of the three adults hold the opinion that there will be another housing bubble in the next four to six years?

Given that \(A\) and \(B\) are two mutually exclusive events, find \(P(A\) or \(B\) ) for the following. a. \(P(A)=.71\) and \(P(B)=.03\) b. \(P(A)=.44\) and \(P(B)=.38\)

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