Step-by-step Solution:

1. Identify Given Information:

   • The experiment has 10 equally likely outcomes. Let the sample space be S. So, the total number of outcomes is $n(S) = 10$.
   • A and B are two non-empty events.
   • Event A consists of 4 outcomes, so $n(A) = 4$.
   • Events A and B are independent.

2. Recall the Condition for Independence: Two events A and B are independent if and only if the probability of their intersection is the product of their individual probabilities: $P(A \cap B) = P(A) \cdot P(B)$

3. Calculate the Probabilities:

   • The probability of event A is the ratio of the number of outcomes in A to the total number of outcomes: $P(A) = \frac{n(A)}{n(S)} = \frac{4}{10} = \frac{2}{5}$
   • Let the number of outcomes in event B be $k$. Since B is a non-empty event, $k$ must be an integer such that $1 \le k \le 10$. The probability of event B is: $P(B) = \frac{n(B)}{n(S)} = \frac{k}{10}$
   • Let the number of outcomes in the intersection of A and B be $m$, so $n(A \cap B) = m$. The probability of the intersection is: $P(A \cap B) = \frac{n(A \cap B)}{n(S)} = \frac{m}{10}$
   • The number of outcomes in the intersection, $m$, must be an integer. Also, since $A \cap B$ is a subset of both A and B, we must have $0 \le m \le n(A)$ and $0 \le m \le n(B)$. That is, $0 \le m \le 4$ and $0 \le m \le k$.

4. Apply the Independence Formula: Substitute the probabilities into the independence equation: $\frac{m}{10} = \left( \frac{4}{10} \right) \cdot \left( \frac{k}{10} \right)$ $\frac{m}{10} = \frac{4k}{100}$ $m = \frac{40k}{100} = \frac{4k}{10} = \frac{2k}{5}$

5. Determine Possible Values for k: We have the equation $m = \frac{2k}{5}$.

   • Since $m$ (the number of outcomes in $A \cap B$) must be an integer, $2k$ must be divisible by 5.
   • As 2 and 5 are prime to each other (co-prime), $k$ must be a multiple of 5.
   • We know that $k$ is the number of outcomes in event B, and $1 \le k \le 10$. The multiples of 5 in this range are 5 and 10.

6. Verify the Possible Values of k:

   • Case 1: $k = 5$ If $n(B) = 5$, then the number of outcomes in the intersection is: $m = \frac{2(5)}{5} = 2$ This is a valid integer. We must also check the subset conditions: $m \le n(A) \Rightarrow 2 \le 4$ (True) and $m \le n(B) \Rightarrow 2 \le 5$ (True). So, it is possible for B to have 5 outcomes.

   • Case 2: $k = 10$ If $n(B) = 10$, then the number of outcomes in the intersection is: $m = \frac{2(10)}{5} = 4$ This is a valid integer. We check the subset conditions: $m \le n(A) \Rightarrow 4 \le 4$ (True) and $m \le n(B) \Rightarrow 4 \le 10$ (True). So, it is possible for B to have 10 outcomes.

7. Conclusion: The possible number of outcomes that event B must have is 5 or 10. This corresponds to option D.

Final Answer Selection:

The calculated possible values for the number of outcomes in B are 5 and 10. Comparing this with the given options: A: 2, 4 or 8 B: 3, 6 or 9 C: 4 or 8 D: 5 or 10

The correct option is D.