If a number x is chosen from the numbers 1, 2, 3 and a number 'y' is chosen from the numbers 1, 4, 9, then P(xy

Question

If a number x is chosen from the numbers 1, 2, 3 and a number 'y' is chosen from the numbers 1, 4, 9, then P(xy < 9) is equal to :

Accepted Answer

To determine the probability $ P(xy < 9) $ where $ x $ is chosen from the numbers 1, 2, 3 and $ y $ is chosen from the numbers 1, 4, 9, we need to consider all possible combinations of $ x $ and $ y $ and count the number of combinations where the product $ xy $ is less than 9.

**Step-by-step Explanation:**

1. **List all possible combinations of $ x $ and $ y $:**

- If $ x = 1 $, then $ y $ can be 1, 4, or 9. The products are:
     - $ 1 \times 1 = 1 $
     - $ 1 \times 4 = 4 $
     - $ 1 \times 9 = 9 $

- If $ x = 2 $, then $ y $ can be 1, 4, or 9. The products are:
     - $ 2 \times 1 = 2 $
     - $ 2 \times 4 = 8 $
     - $ 2 \times 9 = 18 $

- If $ x = 3 $, then $ y $ can be 1, 4, or 9. The products are:
     - $ 3 \times 1 = 3 $
     - $ 3 \times 4 = 12 $
     - $ 3 \times 9 = 27 $

2. **Identify the combinations where $ xy < 9 $:**

- From the above calculations, the combinations where $ xy < 9 $ are:
     - $ (1, 1) $ with product 1
     - $ (1, 4) $ with product 4
     - $ (2, 1) $ with product 2
     - $ (2, 4) $ with product 8
     - $ (3, 1) $ with product 3

There are 5 combinations where $ xy < 9 $.

3. **Total number of possible combinations:**

- There are 3 choices for $ x $ and 3 choices for $ y $, resulting in a total of $ 3 \times 3 = 9 $ possible combinations.

4. **Calculate the probability:**

- The probability $ P(xy < 9) $ is the number of favorable outcomes divided by the total number of possible outcomes:
     $$
     P(xy < 9) = \frac{5}{9}
     $$

**Justification for Option 2:**

- The calculation shows that there are 5 favorable outcomes out of 9 possible outcomes, which matches the probability given in Option 2, $\frac{5}{9}$.

**Explanation for why other options are incorrect:**

- **Option 1: $\frac{7}{9}$** suggests there are 7 favorable outcomes, which is incorrect as we identified only 5.
- **Option 3: $\frac{2}{3}$** is equivalent to $\frac{6}{9}$, suggesting 6 favorable outcomes, which is incorrect.
- **Option 4: $\frac{1}{9}$** suggests only 1 favorable outcome, which is incorrect as there are 5.

Thus, the correct answer is indeed Option 2: $\frac{5}{9}$.