The data give the average number of days in May that are clear and cloudy in two cities.
| | Clear | Cloudy | Total |
| :----------------- | :---- | :----- | :---- |
| Sacramento, CA | 17 | 13 | 30 |
| San Antonio, TX | 6 | 24 | 30 |
| Total | 23 | 37 | 60 |
On a day in May, what is the probability that it is cloudy, given that you are in Sacramento? Explain how you found the answer, and round your answer to two decimal places.
A. 0.35, because 13 of the 37 days were cloudy
B. 0.22, because 13 of the 60 days were cloudy
C. 0.80, because 24 of the 30 days were cloudy
D. 0.43, because 13 of the 30 days were cloudy
Answer (2)
The problem asks for the probability of a cloudy day given that you are in Sacramento.
Identify the number of cloudy days in Sacramento (13) and the total number of days in Sacramento (30).
Calculate the conditional probability: 30 13 \approx 0.43$.
Explanation
Understand the problem We are given a table that shows the number of clear and cloudy days in May for Sacramento and San Antonio. We want to find the probability that a day is cloudy, given that we are in Sacramento. This is a conditional probability problem.
Identify the data From the table, we can see that Sacramento has 13 cloudy days and a total of 30 days in May.
Calculate the conditional probability The conditional probability P(Cloudy | Sacramento) is calculated as the number of cloudy days in Sacramento divided by the total number of days in Sacramento. So, we have: P ( Cloudy ∣ Sacramento ) = Total number of days in Sacramento Number of cloudy days in Sacramento = 30 13
Calculate and round the answer Now, we calculate the value of the fraction and round it to two decimal places: 30 13 ≈ 0.4333
Rounding to two decimal places, we get 0.43.
State the final answer Therefore, the probability that a day in May is cloudy, given that you are in Sacramento, is approximately 0.43.
Examples
Conditional probability is used in many real-life scenarios. For example, in medical diagnosis, it helps doctors determine the probability of a patient having a disease given certain symptoms. In marketing, it can be used to predict the likelihood of a customer making a purchase based on their browsing history. In weather forecasting, it helps predict the probability of rain given certain atmospheric conditions.
The probability of a cloudy day in Sacramento is calculated as the number of cloudy days (13) divided by the total days (30), resulting in approximately 0.43. Therefore, the answer is D. 0.43. This means on any given day in May, there is a 43% chance that it will be cloudy in Sacramento.
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