An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

Options

A. 2,4 or 8
B. 3,6 or 9
C. 4 or 8
D. 5 or 10

Answer

To solve the problem, we need to find the number of outcomes in event B such that events A and B are independent. Here are the steps to arrive at the solution:

### Step 1: Understand the Problem
We have an experiment with 10 equally likely outcomes. Event A consists of 4 outcomes. We need to find the number of outcomes in event B (let's denote this as \( x \)) such that A and B are independent.

### Step 2: Define Probability of Events
The probability of event A can be calculated as:
\[ P(A) = \frac{\text{Number of outcomes in A}}{\text{Total outcomes}} = \frac{4}{10} = \frac{2}{5} \]

### Step 3: Define Probability of Event B
Let the number of outcomes in event B be \( x \). Then the probability of event B is:
\[ P(B) = \frac{x}{10} \]

### Step 4: Use the Independence Condition
For events A and B to be independent, the following condition must hold:
\[ P(A \cap B) = P(A) \cdot P(B) \]
We can express \( P(A \cap B) \) as:
\[ P(A \cap B) = \frac{\text{Number of outcomes in } A \cap B}{\text{Total outcomes}} = \frac{n(A \cap B)}{10} \]

### Step 5: Calculate \( P(A) \cdot P(B) \)
Substituting the probabilities we calculated earlier:
\[ P(A) \cdot P(B) = \left(\frac{2}{5}\right) \cdot \left(\frac{x}{10}\right) = \frac{2x}{50} = \frac{x}{25} \]

### Step 6: Set Up the Equation
Now we can set up the equation:
\[ \frac{n(A \cap B)}{10} = \frac{x}{25} \]
To find \( n(A \cap B) \), we know that \( n(A \cap B) \) can be at most the minimum of the outcomes in A and B. Since A has 4 outcomes, the maximum \( n(A \cap B) \) can be is 4.

### Step 7: Solve for \( x \)
Multiplying both sides by 10:
\[ n(A \cap B) = \frac{10x}{25} = \frac{2x}{5} \]
For \( n(A \cap B) \) to be an integer, \( \frac{2x}{5} \) must also be an integer. This means \( 2x \) must be divisible by 5.

### Step 8: Find Possible Values of \( x \)
To find values of \( x \) that satisfy this condition, we can test values from 1 to 10:
- If \( x = 1 \): \( \frac{2 \cdot 1}{5} = \frac{2}{5} \) (not an integer)
- If \( x = 2 \): \( \frac{2 \cdot 2}{5} = \frac{4}{5} \) (not an integer)
- If \( x = 3 \): \( \frac{2 \cdot 3}{5} = \frac{6}{5} \) (not an integer)
- If \( x = 4 \): \( \frac{2 \cdot 4}{5} = \frac{8}{5} \) (not an integer)
- If \( x = 5 \): \( \frac{2 \cdot 5}{5} = 2 \) (integer)
- If \( x = 6 \): \( \frac{2 \cdot 6}{5} = \frac{12}{5} \) (not an integer)
- If \( x = 7 \): \( \frac{2 \cdot 7}{5} = \frac{14}{5} \) (not an integer)
- If \( x = 8 \): \( \frac{2 \cdot 8}{5} = \frac{16}{5} \) (not an integer)
- If \( x = 9 \): \( \frac{2 \cdot 9}{5} = \frac{18}{5} \) (not an integer)
- If \( x = 10 \): \( \frac{2 \cdot 10}{5} = 4 \) (integer)

Thus, the possible values for \( x \) are 5 and 10.

### Conclusion
The number of outcomes that B must have so that A and B are independent is either 5 or 10.

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