1) In an experiment, the probability that event $$ A $$ occurs is $$ \frac{2}{9} $$, the probability that event $$ B $$ occurs is $$ \frac{2}{7} $$, and the probability that event $$ A $$ occurs given that event $$ B $$ occurs is $$ \frac{2}{9} $$. 4) Are $$ A $$ and $$ B $$ independent events? 4) yes

Question

UpStudy AI · Accepted Answer

Yes, events $$ A $$ and $$ B $$ are **independent**.

### Explanation:

Two events $$ A $$ and $$ B $$ are independent if and only if the occurrence of one event does not affect the probability of the other. Mathematically, this is expressed as:

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

Let's verify this with the given probabilities:

- **Given:**
  - $$ P(A) = \frac{2}{9} $$
  - $$ P(B) = \frac{2}{7} $$
  - $$ P(A \mid B) = \frac{2}{9} $$

- **Check for Independence:**
  
  Compare $$ P(A \mid B) $$ with $$ P(A) $$:
  
  $$
  P(A \mid B) = \frac{2}{9} = P(A)
  $$
  
  Since $$ P(A \mid B) = P(A) $$, this confirms that the occurrence of event $$ B $$ does not affect the probability of event $$ A $$. Therefore, events $$ A $$ and $$ B $$ are independent.

- **Alternative Verification:**

Another way to confirm independence is by checking if:

$$
  P(A \cap B) = P(A) \times P(B)
  $$
  
  Calculate $$ P(A \cap B) $$ using the conditional probability:
  
  $$
  P(A \cap B) = P(A \mid B) \times P(B) = \frac{2}{9} \times \frac{2}{7} = \frac{4}{63}
  $$
  
  Now, calculate $$ P(A) \times P(B) $$:
  
  $$
  P(A) \times P(B) = \frac{2}{9} \times \frac{2}{7} = \frac{4}{63}
  $$
  
  Since both calculations yield $$ \frac{4}{63} $$, this further confirms that $$ A $$ and $$ B $$ are independent events.

### Conclusion:

Your answer is correct. Events $$ A $$ and $$ B $$ are independent.
Events $$ A $$ and $$ B $$ are independent.

In an experiment, the probability that event occurs is , the probability that event
occurs is , and the probability that event occurs given that event occurs is .

Are and independent events?

yes

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In an experiment, the probability that event occurs is , the probability that event occurs is , and the probability that event occurs given that event occurs is . Are and independent events? yes