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In an experiment, the probability that event

A

occurs is

7

8

,

the probability that event

B

occurs is

5

6

,

and the probability that events

A

and

B

both occur is

3

4

.

What is the probability that

A

occurs given that

B

occurs?

*Simplify* *any* *fractions.*

We are asked to find the probability that event \( A \) occurs given that event \( B \) occurs. This is denoted as \( P(A | B) \), and we can use the formula for conditional probability:

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

### Given:
- \( P(A) = \frac{7}{8} \)
- \( P(B) = \frac{5}{6} \)
- \( P(A \cap B) = \frac{3}{4} \)

### Step 1: Substitute the known values into the conditional probability formula:

\[
P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{3}{4}}{\frac{5}{6}}
\]

### Step 2: Simplify the fraction:

To divide fractions, multiply by the reciprocal of the denominator:

\[
P(A | B) = \frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20}
\]

### Step 3: Simplify the result:

\[
\frac{18}{20} = \frac{9}{10}
\]

Thus, the probability that event \( A \) occurs given that event \( B \) occurs is:

\[
P(A | B) = \frac{9}{10}
\]

Let me know if you need further details or clarification!

---

Here are 5 related questions you might find useful:
1. What is the probability that event \( B \) occurs given that event \( A \) occurs?
2. How do you find the probability of the union of two events, \( P(A \cup B) \)?
3. What is the complement of the event \( A \) and how would you calculate \( P(\overline{A}) \)?
4. How do you determine if two events are independent?
5. How would the solution change if the events were mutually exclusive?

**Tip:** Always check that probabilities are valid values between 0 and 1 to avoid calculation errors.

In an experiment, the probability that event A occurs is 7/8, the probability that event B occurs is 5/6, and the probability that events A and B both occur is 3/4. What is the probability that A occurs given that B occurs? Simplify any fractions.

Learn how to find the conditional probability of event A occurring given that event B occurs, with P(A) = 7/8, P(B) = 5/6, and P(A ∩ B) = 3/4. This problem involves applying the conditional probability formula P(A | B) = P(A ∩ B) / P(B) and simplifying the result. Suitable for students in Grades 9-12.

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