Problem 4
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 minivan, given that it owns a house.
Step-by-Step Solution
Verified Answer
The probability that a family owns a minivan, given that they own a house, is approximately 0.3846 or 38.46%.
1Step 1: Write down the conditional probability formula
The conditional probability formula is:
P(A|B) = P(A ∩ B) / P(B)
In this case, event A is the family owning a minivan (M) and event B is the family owning a house (H). So, we'll find P(M|H).
2Step 2: Substitute the given probabilities into the formula
The probabilities we need to substitute are:
- P(H ∩ M) = 0.25
- P(H) = 0.65
So the formula becomes:
P(M|H) = P(M ∩ H) / P(H)
P(M|H) = 0.25 / 0.65
3Step 3: Solve for P(M|H)
Now we simply need to solve the equation:
P(M|H) = 0.25 / 0.65
P(M|H) ≈ 0.3846
So the probability that a family owns a minivan, given that they own a house, is approximately 0.3846 or 38.46%.
Key Concepts
Conditional Probability FormulaProbability with ApplicationsProbability Intersection
Conditional Probability Formula
Understanding the concept of conditional probability is essential to grasp the different scenarios that can arise in probability theory. It refers to the probability of an event occurring, given that another event has already occurred.
For instance, let's consider the scenario provided in our exercise. Families in a town might own a minivan, a house, or both. The conditional probability formula, which is written as \( P(A|B) \), where \( A \) is the event we're interested in and \( B \) is the condition, translates to the probability of event \( A \) given that \( B \) has occurred. The mathematical expression for this is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(A \cap B) \) represents the probability that both events happen together—the intersection—which will be discussed later.
The power of conditional probability lies in its flexibility. It is widely used in fields like finance to estimate the risk of investments given certain economic conditions, and in medicine to determine the likelihood of a health response given a patient's condition. In each application, understanding the dependencies between events is crucial for accurate predictions and decisions.
For instance, let's consider the scenario provided in our exercise. Families in a town might own a minivan, a house, or both. The conditional probability formula, which is written as \( P(A|B) \), where \( A \) is the event we're interested in and \( B \) is the condition, translates to the probability of event \( A \) given that \( B \) has occurred. The mathematical expression for this is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(A \cap B) \) represents the probability that both events happen together—the intersection—which will be discussed later.
The power of conditional probability lies in its flexibility. It is widely used in fields like finance to estimate the risk of investments given certain economic conditions, and in medicine to determine the likelihood of a health response given a patient's condition. In each application, understanding the dependencies between events is crucial for accurate predictions and decisions.
Probability with Applications
Probability isn't just a theoretical concept; it has practical applications across various fields, from simple daily decisions to complex scientific research.
In the exercise, we consider a real-world application where a town's demographic statistic is used to calculate a particular probability. This type of analysis goes well beyond textbook exercises and is a staple in fields such as market research, where businesses might want to know the likelihood of a consumer purchasing one product if they have bought another.
The educational system benefits from understanding probability as well. Curriculums can be developed based on the probability of certain learning outcomes, which helps in tailoring educational content to diverse groups of students. Moreover, by employing conditional probability, students better prepare for real-world scenarios where decisions are informed by prior outcomes or conditions.
In the exercise, we consider a real-world application where a town's demographic statistic is used to calculate a particular probability. This type of analysis goes well beyond textbook exercises and is a staple in fields such as market research, where businesses might want to know the likelihood of a consumer purchasing one product if they have bought another.
The educational system benefits from understanding probability as well. Curriculums can be developed based on the probability of certain learning outcomes, which helps in tailoring educational content to diverse groups of students. Moreover, by employing conditional probability, students better prepare for real-world scenarios where decisions are informed by prior outcomes or conditions.
Probability Intersection
The intersection of events, symbolized by \( \cap \), is crucial in understanding joint occurrences within probability. It defines the scenario where both events happen at the same time.
Consider a Venn diagram with two overlapping circles, where one circle represents families that own a house, and the other represents families that own a minivan. The overlapping area is the intersection, which signifies families owning both a house and a minivan.
In our exercise, the probability of this intersection, \( P(H \cap M) = 0.25 \), is key to solving the conditional probability of a family owning a minivan given that they own a house. Intersections often come into play in statistical analysis, medical diagnostics, and even in understanding traffic flow, making it an integral concept for students to master in order to appreciate the breadth of probability's applications.
Consider a Venn diagram with two overlapping circles, where one circle represents families that own a house, and the other represents families that own a minivan. The overlapping area is the intersection, which signifies families owning both a house and a minivan.
In our exercise, the probability of this intersection, \( P(H \cap M) = 0.25 \), is key to solving the conditional probability of a family owning a minivan given that they own a house. Intersections often come into play in statistical analysis, medical diagnostics, and even in understanding traffic flow, making it an integral concept for students to master in order to appreciate the breadth of probability's applications.
Other exercises in this chapter
Problem 3
Find the coefficient of each. $$x^{2} y^{6} \text { in the expansion of } (2 x+y)^{8}$$
View solution Problem 4
Find the number of positive integers \(\leq 1000\) and \(n\) ot divisible by: 5 or 6
View solution Problem 4
A card is drawn at random from a standard deck of cards. Find the probability of obtaining: A club or a diamond.
View solution Problem 4
Find the coefficient of each. \(x^{4} y^{5}\) in the expansion of \((2 x-3 y)^{9}\)
View solution