An urn contains 11 balls numbered from 1 to 11. if a ball is selected at random, what is the probability of having a ball with a number which is multiple of either 2 or 3 ?

Answer

To find the probability of selecting a ball from the urn that has a number which is a multiple of either 2 or 3, we can follow these steps:

### Step 1: Identify the total number of balls
The urn contains 11 balls, numbered from 1 to 11. Therefore, the total number of outcomes (total balls) is:
\[ N = 11 \]

### Step 2: Identify the favorable outcomes
Next, we need to find the numbers that are multiples of either 2 or 3 within the range of 1 to 11.

**Multiples of 2:**
- The multiples of 2 from 1 to 11 are: 2, 4, 6, 8, 10.
- This gives us 5 favorable outcomes.

**Multiples of 3:**
- The multiples of 3 from 1 to 11 are: 3, 6, 9.
- This gives us 3 favorable outcomes.

### Step 3: Account for overlap
The number 6 is a multiple of both 2 and 3, so we need to ensure we do not double-count it.

Now, we can use the principle of inclusion-exclusion to find the total number of favorable outcomes:
- Total multiples of 2 = 5
- Total multiples of 3 = 3
- Overlap (multiples of both 2 and 3) = 1 (which is 6)

Using the inclusion-exclusion principle:
\[ \text{Total favorable outcomes} = (\text{Multiples of 2}) + (\text{Multiples of 3}) - (\text{Overlap}) \]
\[ = 5 + 3 - 1 = 7 \]

### Step 4: Calculate the probability
The probability \( P \) of selecting a ball that is a multiple of either 2 or 3 is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{11} \]

### Final Answer
The probability of selecting a ball with a number that is a multiple of either 2 or 3 is:
\[ \frac{7}{11} \]

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