Math Problem Statement

or

Watch a video

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.

Solution

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

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

Given:

  • P(A)=78P(A) = \frac{7}{8}
  • P(B)=56P(B) = \frac{5}{6}
  • P(AB)=34P(A \cap B) = \frac{3}{4}

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

P(AB)=P(AB)P(B)=3456P(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(AB)=34×65=3×64×5=1820P(A | B) = \frac{3}{4} \times \frac{6}{5} = \frac{3 \times 6}{4 \times 5} = \frac{18}{20}

Step 3: Simplify the result:

1820=910\frac{18}{20} = \frac{9}{10}

Thus, the probability that event AA occurs given that event BB occurs is:

P(AB)=910P(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 BB occurs given that event AA occurs?
  2. How do you find the probability of the union of two events, P(AB)P(A \cup B)?
  3. What is the complement of the event AA and how would you calculate P(A)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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Probability
Conditional Probability

Formulas

P(A | B) = P(A ∩ B) / P(B)

Theorems

Conditional Probability Theorem

Suitable Grade Level

Grades 9-12