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 of finding the probability that \( p^2 \geq 4q \) when \( p \) and \( q \) are chosen randomly from the set \( \{1, 2, 3, 4\} \) with replacement, we can follow these steps:

### Step 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 outcomes (sample space) is:
\[ \text{Total outcomes} = 4 \times 4 = 16 \]

### Step 2: Determine the Condition
We need to find pairs \( (p, q) \) such that:
\[ p^2 \geq 4q \]
This can be rearranged to:
\[ \frac{p^2}{4} \geq q \]
This means for each value of \( p \), we need to find the corresponding values of \( q \) that satisfy this inequality.

### Step 3: Evaluate Each Possible Value of \( p \)

- **If \( p = 1 \)**:
\[ 1^2 \geq 4q \implies 1 \geq 4q \implies q \leq \frac{1}{4} \]
No valid \( q \) in the set.

- **If \( p = 2 \)**:
\[ 2^2 \geq 4q \implies 4 \geq 4q \implies q \leq 1 \]
Valid \( q \): \( 1 \) (1 case).

- **If \( p = 3 \)**:
\[ 3^2 \geq 4q \implies 9 \geq 4q \implies q \leq \frac{9}{4} = 2.25 \]
Valid \( q \): \( 1, 2 \) (2 cases).

- **If \( p = 4 \)**:
\[ 4^2 \geq 4q \implies 16 \geq 4q \implies q \leq 4 \]
Valid \( q \): \( 1, 2, 3, 4 \) (4 cases).

### Step 4: Count the Favorable Outcomes
Now, we can count the total number of favorable outcomes:
- For \( p = 1 \): 0 cases
- For \( p = 2 \): 1 case
- For \( p = 3 \): 2 cases
- For \( p = 4 \): 4 cases

Adding these up gives:
\[ 0 + 1 + 2 + 4 = 7 \text{ favorable outcomes} \]

### Step 5: Calculate the Probability
The probability \( P \) that \( p^2 \geq 4q \) is given by the ratio of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{16} \]

### Final Answer
Thus, the required probability is:
\[ \frac{7}{16} \]