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 step by step, we need to determine the number of outcomes in event B such that events A and B are independent.

### Step 1: Define the probabilities
Given that the total number of outcomes in the experiment is 10, we can define the probabilities of events A and B.

- Let the number of outcomes in event A be \( |A| = 4 \).
- Therefore, the probability of event A is:
\[ P(A) = \frac{|A|}{\text{Total Outcomes}} = \frac{4}{10} = \frac{2}{5} \]

### Step 2: Define the number of outcomes in event B
Let the number of outcomes in event B be \( |B| = p \). The probability of event B is:
\[ P(B) = \frac{|B|}{\text{Total Outcomes}} = \frac{p}{10} \]

### Step 3: Use the independence condition
For events A and B to be independent, the following condition must hold:
\[ P(A \cap B) = P(A) \times P(B) \]

### Step 4: Calculate \( P(A \cap B) \)
The probability of the intersection of A and B can be expressed as:
\[ P(A \cap B) = \frac{|A \cap B|}{\text{Total Outcomes}} = \frac{|A \cap B|}{10} \]
We also know that:
\[ P(A) \times P(B) = \left(\frac{2}{5}\right) \times \left(\frac{p}{10}\right) = \frac{2p}{50} = \frac{p}{25} \]

### Step 5: Set the equations equal
Equating the two expressions for \( P(A \cap B) \):
\[ \frac{|A \cap B|}{10} = \frac{p}{25} \]

### Step 6: Solve for \( |A \cap B| \)
From the equation above, we can express \( |A \cap B| \):
\[ |A \cap B| = \frac{10p}{25} = \frac{2p}{5} \]

### Step 7: Determine conditions for \( |A \cap B| \)
Since \( |A \cap B| \) must be a non-negative integer, \( \frac{2p}{5} \) must also be an integer. This means that \( p \) must be such that \( 2p \) is divisible by 5.

### Step 8: Find possible values for \( p \)
To satisfy the condition \( \frac{2p}{5} \) being an integer, \( p \) must be a multiple of 5. The possible values of \( p \) (given that \( p \) must be less than or equal to 10) are:
- \( p = 5 \)
- \( p = 10 \)

### Conclusion
Thus, the number of outcomes that event B must have for A and B to be independent is either 5 or 10.