Multiplication Rule: Dependent Events: Videos & Practice Problems

Understanding the probability of events is crucial in both theoretical and practical applications. When dealing with two independent events, the probability can be calculated simply by multiplying their individual probabilities. For instance, if you have a bag containing six marbles—four blue and two red—the probability of drawing a blue marble followed by a red marble, with replacement, is calculated as follows:

Let \( P(\text{Blue}) = \frac{4}{6} \) and \( P(\text{Red}) = \frac{2}{6} \). The combined probability is:

\[ P(\text{Blue and Red}) = P(\text{Blue}) \times P(\text{Red}) = \frac{4}{6} \times \frac{2}{6} = \frac{8}{36} = \frac{2}{9} \]

However, in many real-world scenarios, events are dependent, meaning the occurrence of one event affects the probability of the other. For example, if you draw a blue marble and do not replace it, the total number of marbles in the bag decreases, thus altering the probabilities for subsequent draws. In this case, after drawing a blue marble, the bag now contains five marbles—three blue and two red. The probability of drawing a red marble after removing a blue marble is:

Let \( P(\text{Red | Blue}) = \frac{2}{5} \). The probability of drawing a blue marble first and then a red marble is calculated as:

\[ P(\text{Blue and Red}) = P(\text{Blue}) \times P(\text{Red | Blue}) = \frac{4}{6} \times \frac{2}{5} = \frac{8}{30} = \frac{4}{15} \]

This approach highlights the concept of conditional probability, which is the probability of an event occurring given that another event has already occurred. The notation for conditional probability is expressed as \( P(B | A) \), meaning the probability of event B occurring given that event A has occurred.

To summarize, when calculating the probability of two dependent events, the formula is:

\[ P(A \text{ and } B) = P(A) \times P(B | A) \]

This method allows for the accurate assessment of probabilities in scenarios where events influence one another. Practicing with various examples will enhance your understanding of dependent probabilities and their applications.

In this scenario, we are analyzing the probability that a customer service representative receives a bonus based on customer reviews. To earn the bonus, two conditions must be met: the customer must leave a positive review, and that review must mention the representative by name. We will denote these events as follows:

Let event A be the event that a customer leaves a positive review, and event B be the event that the review mentions the representative by name. We are interested in finding the joint probability of both events occurring, represented as \( P(A \cap B) \).

From the problem, we know the following probabilities:

• The probability of a customer leaving a positive review, \( P(A) \), is 25%, or \( P(A) = 0.25 \).
• The probability that a positive review mentions the representative by name, given that it is a positive review, \( P(B|A) \), is 10%, or \( P(B|A) = 0.10 \).

To find the joint probability \( P(A \cap B) \), we can use the formula:

\( P(A \cap B) = P(A) \times P(B|A) \)

Substituting the known values into the equation gives us:

\( P(A \cap B) = 0.25 \times 0.10 = 0.025 \)

This result indicates that there is a 2.5% probability that a randomly selected customer will leave a positive review that mentions the representative by name, thereby qualifying her for the bonus.

Understanding the concept of conditional probability is essential in statistics, particularly when dealing with dependent events. Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. The formula for calculating conditional probability is expressed as:

P(B|A) = \frac{P(A \cap B)}{P(A)}

In this equation, P(B|A) represents the probability of event B occurring given that event A has occurred. P(A ∩ B) is the probability that both events A and B occur, while P(A) is the probability of event A occurring. This rearrangement of the formula allows us to focus on finding the conditional probability, which is often the desired outcome in real-world problems.

To illustrate this, consider a scenario involving a survey of 40 students in a statistics class, where the majors are distributed as follows: 20 students have a math major, 15 have a science major, 10 have a business major, and 8 have a double major in math and science. If we want to find the probability that a randomly selected student has a math major given that they have a science major, we can identify our events:

Let event A be "the student has a science major" and event B be "the student has a math major." We can then calculate the necessary probabilities:

1. The probability of event A (science major) is:

P(A) = \frac{15}{40}

2. The probability of both events A and B occurring (students with both majors) is:

P(A ∩ B) = \frac{8}{40}

Now, substituting these values into the conditional probability formula gives us:

P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{8}{40}}{\frac{15}{40}} = \frac{8}{15}

This result indicates that the probability of a student having a math major, given that they have a science major, is \(\frac{8}{15}\). This example highlights the practical application of the conditional probability rule, demonstrating how to effectively analyze and interpret data in a statistical context.

In this scenario, we are analyzing the hiring process at a company that has openings for a software developer and a data analyst. A total of 10 candidates applied, with 6 for the software developer position and 4 for the data analyst position. Among these applicants, 4 software developer candidates and 2 data analyst candidates hold a degree in computer science. Our goal is to determine the probability that a randomly selected candidate with a computer science degree applied for the software developer position.

This situation involves conditional probability, where we are interested in the probability of one event occurring given that another event has already occurred. Here, we define two events: event A represents having a degree in computer science, and event B represents applying for the software developer position. We seek to find the probability of event B given event A, denoted as P(B|A).

The formula for conditional probability is:

P(B|A) = \frac{P(A \cap B)}{P(A)}

To apply this formula, we first need to calculate the probabilities of events A and B:

1. **Calculating P(A)**: This is the probability that a candidate has a computer science degree. We know that there are 6 candidates with a computer science degree (4 from the software developer pool and 2 from the data analyst pool) out of a total of 10 applicants. Thus,

P(A) = \frac{6}{10} = 0.6

2. **Calculating P(A ∩ B)**: This is the probability that a candidate has a computer science degree and applied for the software developer position. Since 4 candidates who applied for the software developer position have a computer science degree, we find that:

P(A \cap B) = \frac{4}{10} = 0.4

Now, we can substitute these values into the conditional probability formula:

P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.4}{0.6} = \frac{2}{3} \approx 0.67

Therefore, the probability that a candidate applied for the software developer position given that they have a computer science degree is approximately two-thirds, or 0.67. This indicates a strong likelihood that a candidate with a computer science degree is applying for the software developer role.

In this scenario, an insurance company is evaluating whether to increase home insurance prices for pet owners based on the likelihood of them filing claims. The company has determined that 5% of their clients who are pet owners have filed claims, while 60% of their clients own pets. The key factor in their decision is the conditional probability of filing a claim given that a client is a pet owner.

To analyze this, we define two events: let event A represent being a pet owner, and event B represent filing a claim. The company is interested in calculating the conditional probability \( P(B|A) \), which is the probability of filing a claim given that the client is a pet owner. This can be calculated using the formula:

\( P(B|A) = \frac{P(A \cap B)}{P(A)} \)

Here, \( P(A \cap B) \) is the probability that a client is both a pet owner and has filed a claim, which is given as 5% or \( 0.05 \). The probability \( P(A) \), which is the probability that a client is a pet owner, is 60% or \( 0.60 \).

Substituting these values into the formula gives:

\( P(B|A) = \frac{0.05}{0.60} \approx 0.0833 \)

This result indicates that the probability of a pet owner filing a claim is approximately 8.33%, which exceeds the threshold of 8% set by the insurance company. Therefore, the company concludes that they will raise the insurance prices for pet owners.