Question

1.If P(A and B) = 0.14, and P(A) = 0.35, What is
P(B/A)?

2.Given P(B) = 0.45, and P(A/B) = 0.35, What is P(A
and B)?

3.What does P(A/B) mean?
A. Both A and B are dependent
B. A is true given that B is true
C. B is true given that A is true
D. Both A and B are independent

4. Independent or Dependent:
Given P(A) 3/10, P=(B)2/5 , and P(A and B) = 3/25

5. Independent or Dependent:
Given P(A) = 7/20 , P(B) = 1/4, and P(A/B) = 21/100

any help would be greatly needed if possible can you explain the answers as well.

Answer 1

Using probability concepts, it is found that:

1. P(B|A) = 0.4.

2. $P(A \cap B) = 0.1575$

3. B. A is true given that B is true

4. Since $P(A \cap B) = P(A)P(B)$, they are dependent.

5. Since $P(A \cap B) \neq P(A)P(B)$, they are independent.

Conditional Probability

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

In which

• P(B|A) is the probability of event B happening, given that A happened.

• $P(A \cap B)$ is the probability of both A and B happening.

• P(A) is the probability of A happening.

Item 1:

Applying conditional probability:

$P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.14}{0.35} = 0.4$

Hence, P(B|A) = 0.4.

Item 2:

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

$0.35 = \frac{P(A \cap B)}{0.45}$

$P(A \cap B) = 0.35(0.45)$

$P(A \cap B) = 0.1575$

Item 3:

A is true given that B is true, hence, option B.

Item 4:

Two events, A and B, are dependent if:

$P(A \cap B) = P(A)P(B)$

In this problem:

$P(A \cap B) = \frac{3}{25}$

$P(A)P(B) = \frac{3}{10} \times \frac{2}{5} = \frac{6}{50} = \frac{3}{25}$

Since $P(A \cap B) = P(A)P(B)$, they are dependent.

Item 5:

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

$P(A \cap B) = P(B)P(A|B)$

$P(A \cap B) = \frac{1}{4} \times \frac{21}{100}$

$P(A \cap B) = \frac{21}{400}$

$P(A)P(B) = \frac{7}{20} \times \frac{1}{4} = \frac{7}{80}$

Since $P(A \cap B) \neq P(A)P(B)$, they are independent.

To learn more about probability concepts, you can take a look at https://brainly.com/question/14398287