Problem 61 - College Algebra with Modeling and Visualization

Question

Conditional Probability and Dependent Events The probability of a day being cloudy is \(30 \%,\) and the probability of it being cloudy and windy is \(12 \% .\) Given that the day is cloudy, what is the probability that it will be windy?

Step-by-Step Solution

Answer

The probability that it will be windy given it's cloudy is 40%.

Step 1: Identify the Known Probabilities

We're given two probabilities: the probability of a day being cloudy, denoted as \( P(C) = 0.30 \), and the probability of it being both cloudy and windy, denoted as \( P(C \, \text{and} \, W) = 0.12 \).

Step 2: Apply Conditional Probability Formula

To find the probability of a day being windy given it's cloudy, \( P(W|C) \), use the formula for conditional probability: \[ P(W|C) = \frac{P(C \text{ and } W)}{P(C)} \]

Step 3: Calculate the Conditional Probability

Substitute the known probabilities into the formula: \[ P(W|C) = \frac{0.12}{0.30} \].

Step 4: Simplify the Expression

Divide 0.12 by 0.30 to find \( P(W|C) \): \[ P(W|C) = \frac{0.12}{0.30} = 0.4 \].

Step 5: Conclusion

The probability that it will be windy given that the day is cloudy is 0.4 or 40%.

Key Concepts

Dependent Events

In probability, events can be independent or dependent. Dependent events are those where the outcome or probability of one event affects the outcome or probability of another. When we talk about days being cloudy and windy, these can be dependent events. If one considers the climate of a region, often, whether it's cloudy may affect the likelihood of it being windy. Understanding the dependency between events helps in determining accurate probabilities. For example, if a day is cloudy which might indicate a low-pressure weather system, it could increase the chance that it's also windy. This makes cloudy and windy days dependent if one event influences the occurrence of the other.

Probability Formula

The probability formula helps calculate the chance that a particular event will happen. When dealing with dependent events, such as determining the probability of a windy day given that it is cloudy, we use the conditional probability formula. This is expressed as:

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

Here, \( P(A|B) \) is the probability of event A happening given that B has occurred, \( P(A \text{ and } B) \) is the probability of both events A and B occurring together, and \( P(B) \) is the probability of event B. This formula allows us to adjust the probability based on additional known information, which is crucial for understanding events that are related.

Cloudy and Windy Days

Cloudy days often signify specific atmospheric conditions that could influence whether it will also be windy. These are two particular weather conditions whose likelihoods can be interconnected, captured in probability terms. The given problem states that 30% of days are cloudy, and cloudy and windy days make up 12% of the days. This indicates that not every cloudy day will be windy, but there is a significant overlap where both conditions happen. This kind of real-life scenario illustrates how contextual data helps us assess probability, helping us see that while cloudy days are not universally windy, there's a portion where both occur.

Probability Calculation

Calculating conditional probability involves inserting known values into the formula and solving from there. Given the problem, we know:

• The probability of a cloudy day, \( P(C) = 0.30 \)
• The probability of a day being both cloudy and windy, \( P(C \text{ and } W) = 0.12 \)

Substitute these into the conditional probability formula:

• \[ P(W|C) = \frac{P(C \text{ and } W)}{P(C)} = \frac{0.12}{0.30} \]

Simplify this to get:

• \[ P(W|C) = 0.4 \]

This calculation means that there is a 40% chance it will be windy on a cloudy day. Such calculations are very useful in weather predictions and other real-world applications where understanding conditional scenarios is important.