Problem 3 - Discrete Mathematics with Applications

Question

It is found that 65\(\%\) of the families in a town own a house, 25\(\%\) own a house and a minivan, and 40\(\%\) own a minivan. Find the probability that a family selected at random owns each of the following. A house, given that it owns a minivan.

Step-by-Step Solution

Answer

The probability that a randomly selected family owns a house given that it owns a minivan is 0.625 or 62.5%.

Step 1: Understanding the problem

We know the following probabilities: - The probability of a family owning a house is 65%. - The probability of a family owning a house and a minivan is 25%. - The probability of a family owning a minivan is 40%. We are asked to find the probability that a randomly selected family owns a house given that it owns a minivan.

Step 2: Write down the conditional probability formula

For any two events A and B, the conditional probability of event A happening given event B has occurred is: P(A|B) = \(\frac{P(A \cap B)}{P(B)}\) In our case, the event A represents owning a house, and the event B represents owning a minivan.

Step 3: Substitute the values and calculate the probability

Now we will substitute the given probabilities into the conditional probability formula: P(A|B) = P(House | Minivan) = \(\frac{P(House \cap Minivan)}{P(Minivan)}\) Where: - P(House | Minivan) is the probability of a family owning a house given that they own a minivan. - P(House \(\cap\) Minivan) is the probability of a family owning a house and a minivan which is 25%. - P(Minivan) is the probability of a family owning a minivan which is 40%. So, the formula becomes: P(House | Minivan) = \(\frac{0.25}{0.40}\)

Step 4: Calculate the result

Now we will calculate the probability: P(House | Minivan) = \(\frac{0.25}{0.40}\) = 0.625

Step 5: Interpret the result

The probability that a randomly selected family owns a house given that it owns a minivan is 0.625 or 62.5%. So, there is a 62.5% chance that a family in this town owns a house if we already know they own a minivan.

Key Concepts

Probability Theory

Probability theory is all about quantifying uncertainty. It involves the use of mathematics to model and analyze scenarios where outcomes are uncertain. In everyday life, uncertainty is everywhere, from weather forecasts to stock market predictions. Probability theory provides us with the tools to make informed predictions about such uncertain events.

In probability, we often denote the likelihood of an event occurring as a value between 0 and 1. A probability of 0 indicates an impossibility, while a probability of 1 indicates certainty. Here are a few key elements:

• Event: An outcome or a set of outcomes.
• Sample space: All possible outcomes.
• Probability of an event (P): A measure of the likelihood that an event will occur.

For instance, in the exercise, the probability of a family owning a house (65%) and a minivan (40%) are events in our sample space. Probability theory helps us understand how likely these events are to happen.

Bayes' Theorem

Bayes' theorem is a cornerstone of probability theory and statistics that allows us to update the probability of a hypothesis based on new evidence or information. It connects conditional probabilities with marginal probabilities, offering a mathematical way to revise our beliefs in light of new data.

Bayes' theorem can be particularly handy when dealing with complex problems involving dependent events. The general formula for Bayes' theorem is:\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]Where:

• \(P(A|B)\) is the probability of hypothesis \(A\) given the data \(B\).
• \(P(B|A)\) is the probability of data \(B\) given that \(A\) is true.
• \(P(A)\) is the prior probability of \(A\).
• \(P(B)\) is the marginal probability of \(B\).

In the exercise, the conditional probability \(P(\text{House} | \text{Minivan})\) is calculated using this theorem, given that the various probabilities are already known. This method allows us to deduce the likelihood of owning a house when it's known that a minivan is owned.

Probability and Statistics

Probability and statistics are intertwined fields that help us make sense of the world by interpreting data and uncertainty. Probability provides the theoretical foundation, while statistics refers to the method of analyzing and interpreting data to make sense of real-world phenomena.

In probability and statistics:

• Probability deals with predicting the likelihood of future events.
• Statistics involves the analysis of the frequency of past events.

The exercise involves probability by calculating how likely a family is to own a house if they already own a minivan, based on known statistics of ownership in a town. This involves using the data — the frequencies of families owning houses, minivans, or both — to make a meaningful prediction.

Understanding these concepts allows you to make informed decisions based on data, whether defining the reliability of a survey, conducting medical tests, or evaluating investment risks. In essence, probability and statistics provide a framework for making sound decisions under uncertainty.