A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Jet A be the event “number obtained is even” and. B be B the event “number obtained is red”. Find ifA and B are independent events.

Answer

To determine whether events A and B are independent, we need to follow these steps:

### Step 1: Define the Events
- Let event A be the event that the number obtained is even.
- Let event B be the event that the number obtained is red.

### Step 2: Identify the Outcomes
The die has the following faces:
- Red faces: 1, 2, 3
- Green faces: 4, 5, 6

### Step 3: Determine the Outcomes for Event A
Event A (even numbers) includes:
- Even numbers on the die: 2, 4, 6
- Therefore, the favorable outcomes for event A are {2, 4, 6}.

### Step 4: Calculate Probability of Event A
The total number of outcomes when tossing the die is 6.
- The number of favorable outcomes for event A (even numbers) is 3 (2, 4, 6).
- Thus, the probability of event A is:
\[ P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]

### Step 5: Determine the Outcomes for Event B
Event B (red numbers) includes:
- Red numbers on the die: 1, 2, 3
- Therefore, the favorable outcomes for event B are {1, 2, 3}.

### Step 6: Calculate Probability of Event B
- The number of favorable outcomes for event B (red numbers) is 3 (1, 2, 3).
- Thus, the probability of event B is:
\[ P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]

### Step 7: Determine the Intersection of Events A and B
The intersection of events A and B (A ∩ B) includes outcomes that are both even and red:
- The common outcome in both events is {2}.
- Therefore, the favorable outcome for A ∩ B is {2}.

### Step 8: Calculate Probability of A ∩ B
- The number of favorable outcomes for A ∩ B is 1 (only the number 2).
- Thus, the probability of A ∩ B is:
\[ P(A \cap B) = \frac{\text{Number of favorable outcomes for A ∩ B}}{\text{Total outcomes}} = \frac{1}{6} \]

### Step 9: Check for Independence
Two events A and B are independent if:
\[ P(A \cap B) = P(A) \times P(B) \]

Calculating \(P(A) \times P(B)\):
\[ P(A) \times P(B) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{4} \]

### Step 10: Compare the Probabilities
Now we compare:
- \(P(A \cap B) = \frac{1}{6}\)
- \(P(A) \times P(B) = \frac{1}{4}\)

Since \(\frac{1}{6} \neq \frac{1}{4}\), we conclude that:
\[ P(A \cap B) \neq P(A) \times P(B) \]

### Conclusion
Events A and B are not independent.

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