On choosing a number x from the numbers 1, 2, 3 and a number y from the numbers 1, 4, 9, the probability of P(xy < 9) is:

Options

A. 0.55555555555556
B. 0.11111111111111
C. 0.44444444444444
D. 0.33333333333333

Answer

To solve the problem of finding the probability that the product \( xy < 9 \) when choosing \( x \) from the set \( \{1, 2, 3\} \) and \( y \) from the set \( \{1, 4, 9\} \), we will follow these steps:

### Step 1: Identify the Sample Space
The sample space consists of all possible pairs \( (x, y) \) that can be formed by choosing \( x \) from \( \{1, 2, 3\} \) and \( y \) from \( \{1, 4, 9\} \).

- Possible pairs:
- For \( x = 1 \): \( (1, 1), (1, 4), (1, 9) \)
- For \( x = 2 \): \( (2, 1), (2, 4), (2, 9) \)
- For \( x = 3 \): \( (3, 1), (3, 4), (3, 9) \)

Thus, the complete sample space is:
\[ \{(1, 1), (1, 4), (1, 9), (2, 1), (2, 4), (2, 9), (3, 1), (3, 4), (3, 9)\} \]
The total number of outcomes in the sample space is \( 9 \).

### Step 2: Determine Favorable Outcomes
Next, we need to find the pairs where the product \( xy < 9 \).

- For \( (1, 1) \): \( 1 \times 1 = 1 < 9 \) (favorable)
- For \( (1, 4) \): \( 1 \times 4 = 4 < 9 \) (favorable)
- For \( (1, 9) \): \( 1 \times 9 = 9 \) (not favorable)
- For \( (2, 1) \): \( 2 \times 1 = 2 < 9 \) (favorable)
- For \( (2, 4) \): \( 2 \times 4 = 8 < 9 \) (favorable)
- For \( (2, 9) \): \( 2 \times 9 = 18 \) (not favorable)
- For \( (3, 1) \): \( 3 \times 1 = 3 < 9 \) (favorable)
- For \( (3, 4) \): \( 3 \times 4 = 12 \) (not favorable)
- For \( (3, 9) \): \( 3 \times 9 = 27 \) (not favorable)

The favorable outcomes are:
\[ \{(1, 1), (1, 4), (2, 1), (2, 4), (3, 1)\} \]
The total number of favorable outcomes is \( 5 \).

### Step 3: Calculate the Probability
The probability \( P(xy < 9) \) is given by the formula:
\[ P(xy < 9) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{9} \]

### Final Answer
Thus, the probability that the product \( xy < 9 \) is:
\[ \boxed{\frac{5}{9}} \]