During a year ending July 31 of that year, there were approximately 6.2 million sales of existing homes in the United States, of which 1.6 million were sold in the West. During July of that year there were a total of 590,000 existing homes sold in the United States, of which 120,000 were sold in the West. (Round your answers to two decimal places.)
(a)
Find the probability that a home sale in the year ending July 31 of that year, took place in the West, given that the home was sold during July of that year.
(b)
Find the probability that a home sale in the year ending July 31 of that year, took place in July of that year, given that it took place in the West.


We can solve both questions using conditional probability formulas. Let's break it down step by step.

### Definitions
- \( A \): A home sale took place in the West.
- \( B \): A home sale took place in July.

### Given Information:
- Total home sales for the year: 6.2 million
- Total home sales in the West for the year: 1.6 million
- Total home sales in July: 590,000
- Total home sales in the West in July: 120,000

### (a) Find the probability that a home sale took place in the West, given that it was sold during July.
This is \( P(A|B) \), which can be calculated using the conditional probability formula:

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

Where:
- \( P(A \cap B) \) is the probability that the home sale was both in the West and in July.
- \( P(B) \) is the probability that the home sale took place in July.

Let's calculate each part:
1. \( P(A \cap B) \): The probability of a home being sold in the West and in July is the number of homes sold in the West in July divided by the total number of homes sold in the entire year:
   \[
   P(A \cap B) = \frac{120,000}{6,200,000}
   \]

2. \( P(B) \): The probability of a home being sold in July is the total number of homes sold in July divided by the total number of homes sold in the year:
   \[
   P(B) = \frac{590,000}{6,200,000}
   \]

Now we can plug these values into the conditional probability formula for part (a).

### (b) Find the probability that a home sale took place in July, given that it was sold in the West.
This is \( P(B|A) \), which again can be calculated using the conditional probability formula:

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

Where:
- \( P(A \cap B) \) is the same as calculated above.
- \( P(A) \) is the probability that a home sale took place in the West, which is the number of homes sold in the West divided by the total number of homes sold during the year:
  \[
  P(A) = \frac{1,600,000}{6,200,000}
  \]

Now, let's calculate both probabilities.### Results:
- **(a)** The probability that a home sale took place in the West, given that it was sold during July is approximately **0.20** (or 20.34%).
- **(b)** The probability that a home sale took place in July, given that it was sold in the West is approximately **0.075** (or 7.5%).

Would you like further details on how these probabilities were computed or any clarifications?

Here are 5 related questions:
1. How do you interpret conditional probabilities in real-life situations?
2. What does it mean if \( P(A|B) \) and \( P(B|A) \) are different values?
3. How does the total number of sales impact the probabilities?
4. Can conditional probability be used to predict future sales trends?
5. How would you calculate the probability if we had data from multiple regions, not just the West?

**Tip**: Conditional probability often reveals hidden relationships between events, making it a powerful tool for data-driven decision-making!

During a year ending July 31 of that year, there were approximately 6.2 million sales of existing homes in the United States, of which 1.6 million were sold in the West. During July of that year there were a total of 590,000 existing homes sold in the United States, of which 120,000 were sold in the West. (Round your answers to two decimal places.) (a) Find the probability that a home sale in the year ending July 31 of that year, took place in the West, given that the home was sold during July of that year. (b) Find the probability that a home sale in the year ending July 31 of that year, took place in July of that year, given that it took place in the West.

Learn how to calculate conditional probabilities related to U.S. home sales data. Find the probability that a home sale occurred in the West given it was sold in July, and vice versa. Includes key statistics, formulas, and a step-by-step solution. Suitable for Grades 10-12 students.

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