Given That It Was Less Than $80^{\circ} F$ On A Given Day, What Is The Probability That It Also Rained That Day?A. 0.3 B. 0.35 C. 0.65 D. 0.7 The Conditional Relative Frequency Table Was Generated Using Data That Compared The Outside

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Conditional Probability: Understanding the Relationship Between Temperature and Rainfall

Conditional probability is a fundamental concept in mathematics that helps us understand the relationship between two events. In this article, we will explore the concept of conditional probability using a real-world example: the relationship between temperature and rainfall. We will use a conditional relative frequency table to analyze the data and calculate the probability of rainfall on a given day when the temperature is less than 80°F.

What is Conditional Probability?

Conditional probability is a measure of the likelihood of an event occurring given that another event has occurred. It is denoted by the symbol P(A|B) and is read as "the probability of A given B." In other words, it is the probability of event A occurring when event B has already occurred.

The Data

The data used in this analysis was collected from a weather station and consists of two variables: temperature and rainfall. The temperature is measured in degrees Fahrenheit, and the rainfall is recorded as either "yes" or "no." The data is summarized in the following conditional relative frequency table:

Temperature (°F) Rainfall (Yes/No) Frequency
Less than 80 Yes 120
Less than 80 No 180
80 or more Yes 150
80 or more No 250

Calculating the Probability

To calculate the probability of rainfall on a given day when the temperature is less than 80°F, we need to use the formula for conditional probability:

P(Rain|Less than 80°F) = P(Rain and Less than 80°F) / P(Less than 80°F)

We can calculate the numerator and denominator separately using the data from the table.

Numerator: P(Rain and Less than 80°F)

The numerator represents the probability of both events occurring: rainfall and temperature less than 80°F. We can calculate this by dividing the frequency of rainfall on days with temperature less than 80°F by the total frequency of days with temperature less than 80°F.

P(Rain and Less than 80°F) = 120 / (120 + 180) = 120 / 300 = 0.4

Denominator: P(Less than 80°F)

The denominator represents the probability of the event occurring: temperature less than 80°F. We can calculate this by dividing the total frequency of days with temperature less than 80°F by the total frequency of all days.

P(Less than 80°F) = (120 + 180) / (120 + 180 + 150 + 250) = 300 / 620 = 0.484

Conditional Probability

Now that we have calculated the numerator and denominator, we can calculate the conditional probability of rainfall on a given day when the temperature is less than 80°F.

P(Rain|Less than 80°F) = P(Rain and Less than 80°F) / P(Less than 80°F) = 0.4 / 0.484 = 0.823

Conclusion

In this article, we used a conditional relative frequency table to analyze the relationship between temperature and rainfall. We calculated the conditional probability of rainfall on a given day when the temperature is less than 80°F and found that it is approximately 0.823. This means that on a given day when the temperature is less than 80°F, there is an approximately 82.3% chance of rainfall.

Answer

The correct answer is not listed in the options provided. However, based on our calculation, the probability of rainfall on a given day when the temperature is less than 80°F is approximately 0.823.

References

Note

The data used in this analysis is fictional and for illustrative purposes only. The actual data may vary depending on the location and time of year.
Conditional Probability: Q&A

In our previous article, we explored the concept of conditional probability using a real-world example: the relationship between temperature and rainfall. We calculated the conditional probability of rainfall on a given day when the temperature is less than 80°F and found that it is approximately 0.823. In this article, we will answer some frequently asked questions about conditional probability.

Q: What is the difference between conditional probability and regular probability?

A: Conditional probability is a measure of the likelihood of an event occurring given that another event has occurred. Regular probability, on the other hand, is a measure of the likelihood of an event occurring without any conditions. For example, the probability of rolling a 6 on a fair six-sided die is 1/6, but the probability of rolling a 6 given that the die is fair is 1/6.

Q: How do I calculate conditional probability?

A: To calculate conditional probability, you need to use the formula:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the conditional probability of event A given event B, P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.

Q: What is the difference between conditional probability and independence?

A: Conditional probability and independence are related but distinct concepts. Independence means that the occurrence of one event does not affect the probability of another event. Conditional probability, on the other hand, takes into account the occurrence of one event when calculating the probability of another event.

Q: Can conditional probability be used in real-world applications?

A: Yes, conditional probability has many real-world applications, such as:

  • Predicting the likelihood of a patient having a certain disease given their symptoms
  • Calculating the probability of a stock price increasing given the current market conditions
  • Determining the likelihood of a natural disaster occurring given the current weather conditions

Q: How do I interpret the results of a conditional probability calculation?

A: When interpreting the results of a conditional probability calculation, you need to consider the following:

  • The conditional probability is a measure of the likelihood of an event occurring given that another event has occurred.
  • The conditional probability is not the same as the regular probability of the event occurring.
  • The conditional probability can be used to make informed decisions or predictions.

Q: What are some common mistakes to avoid when calculating conditional probability?

A: Some common mistakes to avoid when calculating conditional probability include:

  • Failing to account for the conditional probability formula
  • Using the wrong values for the numerator and denominator
  • Failing to consider the independence of the events

Conclusion

In this article, we answered some frequently asked questions about conditional probability. We discussed the difference between conditional probability and regular probability, how to calculate conditional probability, and how to interpret the results of a conditional probability calculation. We also highlighted some common mistakes to avoid when calculating conditional probability.

References

Note

The information provided in this article is for educational purposes only and is not intended to be used as a substitute for professional advice or guidance.