The problem presented in the image is asking to determine whether the events $A$ and $B$ are dependent or independent.

• Event A: The coin shows "герб" (tails).
• Event B: The dice shows a number divisible by 3.

Analyzing Independence

To check if the two events are independent, we use the standard rule of probability:

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

where:

• $P(A)$ is the probability of event $A$,
• $P(B)$ is the probability of event $B$,
• $P(A \cap B)$ is the probability of both events happening together.

Step 1: Calculating $P(A)$

Since we are tossing a coin, the probability of getting "tails" (event $A$) is:

$P(A) = \frac{1}{2}$

Step 2: Calculating $P(B)$

The dice has 6 sides, and the numbers divisible by 3 are 3 and 6. Therefore, there are two favorable outcomes for event $B$:

$P(B) = \frac{2}{6} = \frac{1}{3}$

Step 3: Calculating $P(A \cap B)$

The coin and dice are tossed simultaneously. The probability of both getting "tails" (event $A$) and a number divisible by 3 on the dice (event $B$) is:

$P(A \cap B) = P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}$

Step 4: Checking Independence

Now, check if $P(A \cap B) = P(A) \cdot P(B)$. Since:

$P(A \cap B) = \frac{1}{6}$

and:

$P(A) \cdot P(B) = \frac{1}{6}$

The two events $A$ and $B$ are independent because the condition for independence holds true.

Conclusion

The correct answer is: Независимы (Independent).

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the formula for conditional probability?
2. How can we calculate the probability of a union of two events?
3. What does it mean if two events are dependent?
4. How would the answer change if the events involved rolling two dice instead of one?
5. Can two mutually exclusive events ever be independent?

Tip: For two events to be independent, knowing the outcome of one event should not affect the probability of the other event occurring.