If two numbers p and q are choosen randomly from the set {1, 2, 3, 4} with replacement, then the probability that `p^(2)ge4q` is equal to

Options

A. `(1)/(4)`
B. `(3)/(16)`
C. `(1)/(2)`
D. `(7)/(16)`

Answer

To solve the problem, we need to find the probability that \( p^2 \geq 4q \) where \( p \) and \( q \) are chosen randomly from the set \( \{1, 2, 3, 4\} \) with replacement.

### Step-by-Step Solution:

1. **Identify the Sample Space**:
Since \( p \) and \( q \) can each take any of the 4 values from the set \( \{1, 2, 3, 4\} \), the total number of possible pairs \( (p, q) \) is:
\[ 4 \times 4 = 16 \]

2. **Evaluate the Condition \( p^2 \geq 4q \)**:
We will check each possible value of \( p \) and see which values of \( q \) satisfy the condition.

- **Case 1**: \( p = 1 \)
\[ 1^2 \geq 4q \implies 1 \geq 4q \implies q \leq \frac{1}{4} \]
No possible values for \( q \) since \( q \) starts from 1.

- **Case 2**: \( p = 2 \)
\[ 2^2 \geq 4q \implies 4 \geq 4q \implies 1 \geq q \]
The only possible value for \( q \) is 1.

- **Case 3**: \( p = 3 \)
\[ 3^2 \geq 4q \implies 9 \geq 4q \implies q \leq \frac{9}{4} = 2.25 \]
Possible values for \( q \) are 1 and 2.

- **Case 4**: \( p = 4 \)
\[ 4^2 \geq 4q \implies 16 \geq 4q \implies 4 \geq q \]
Possible values for \( q \) are 1, 2, 3, and 4.

3. **Count the Favorable Outcomes**:
Now we will count the pairs \( (p, q) \) that satisfy \( p^2 \geq 4q \):
- For \( p = 1 \): 0 pairs
- For \( p = 2 \): 1 pair \( (2, 1) \)
- For \( p = 3 \): 2 pairs \( (3, 1) \) and \( (3, 2) \)
- For \( p = 4 \): 4 pairs \( (4, 1), (4, 2), (4, 3), (4, 4) \)

Total favorable outcomes:
\[ 0 + 1 + 2 + 4 = 7 \]

4. **Calculate the Probability**:
The probability \( P \) that \( p^2 \geq 4q \) is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{16} \]

### Final Answer:
The probability that \( p^2 \geq 4q \) is \( \frac{7}{16} \).