A number x is selected at random from the numbers 1, 2, 3 and 4. Another number y is selected at random from the numbers 1, 4, 9 and 16. Find the probability that product of x and y is less than 16.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the sets of numbers
- The number \( x \) is selected from the set \( \{1, 2, 3, 4\} \).
- The number \( y \) is selected from the set \( \{1, 4, 9, 16\} \).

### Step 2: Determine the total number of outcomes
- The total number of ways to select \( x \) is 4 (since there are 4 options: 1, 2, 3, 4).
- The total number of ways to select \( y \) is also 4 (since there are 4 options: 1, 4, 9, 16).
- Therefore, the total number of outcomes when selecting both \( x \) and \( y \) is:
\[ \text{Total Outcomes} = 4 \times 4 = 16 \]

### Step 3: Find the favorable outcomes where the product \( xy < 16 \)
We will evaluate each possible value of \( x \) and determine which values of \( y \) satisfy the condition \( xy < 16 \).

1. **If \( x = 1 \)**:
- \( y = 1 \): \( 1 \times 1 = 1 < 16 \) (favorable)
- \( y = 4 \): \( 1 \times 4 = 4 < 16 \) (favorable)
- \( y = 9 \): \( 1 \times 9 = 9 < 16 \) (favorable)
- \( y = 16 \): \( 1 \times 16 = 16 \) (not favorable)
- **Favorable outcomes**: 3

2. **If \( x = 2 \)**:
- \( y = 1 \): \( 2 \times 1 = 2 < 16 \) (favorable)
- \( y = 4 \): \( 2 \times 4 = 8 < 16 \) (favorable)
- \( y = 9 \): \( 2 \times 9 = 18 \) (not favorable)
- \( y = 16 \): \( 2 \times 16 = 32 \) (not favorable)
- **Favorable outcomes**: 2

3. **If \( x = 3 \)**:
- \( y = 1 \): \( 3 \times 1 = 3 < 16 \) (favorable)
- \( y = 4 \): \( 3 \times 4 = 12 < 16 \) (favorable)
- \( y = 9 \): \( 3 \times 9 = 27 \) (not favorable)
- \( y = 16 \): \( 3 \times 16 = 48 \) (not favorable)
- **Favorable outcomes**: 2

4. **If \( x = 4 \)**:
- \( y = 1 \): \( 4 \times 1 = 4 < 16 \) (favorable)
- \( y = 4 \): \( 4 \times 4 = 16 \) (not favorable)
- \( y = 9 \): \( 4 \times 9 = 36 \) (not favorable)
- \( y = 16 \): \( 4 \times 16 = 64 \) (not favorable)
- **Favorable outcomes**: 1

### Step 4: Calculate the total number of favorable outcomes
- Total favorable outcomes = \( 3 + 2 + 2 + 1 = 8 \)

### Step 5: Calculate the probability
- The probability \( P \) that the product \( xy < 16 \) is given by the formula:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{8}{16} = \frac{1}{2} \]

### Final Answer
The probability that the product of \( x \) and \( y \) is less than 16 is \( \frac{1}{2} \).

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