Step 1: Understanding the Concept:
We are drawing two numbers $p$ and $q$ with replacement, so all pairs $(p, q)$ are equally likely.
We need to find the total number of possible pairs and the number of pairs that satisfy the given condition $p^2 \ge 4q$.
Step 2: Key Formula or Approach:
Total outcomes = (number of choices for $p$) $\times$ (number of choices for $q$).
Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.
Step 3: Detailed Explanation:
The set of numbers is $S = \{1, 2, 3, 4\}$.
Since numbers are drawn with replacement, $p \in S$ and $q \in S$.
Total number of possible pairs $(p, q) = 4 \times 4 = 16$.
We want to find pairs satisfying the condition $p^2 \ge 4q$. Let's test each possible value of $p$:
- Case 1: If $p = 1$, then $p^2 = 1$. The condition is $1 \ge 4q$.
Since minimum value of $q$ is 1, $4q \ge 4$. Thus, $1 \ge 4$ is false. No $q$ satisfies this. (0 pairs)
- Case 2: If $p = 2$, then $p^2 = 4$. The condition is $4 \ge 4q \Rightarrow 1 \ge q$.
The only value from the set satisfying this is $q = 1$. So, the pair is $(2, 1)$. (1 pair)
- Case 3: If $p = 3$, then $p^2 = 9$. The condition is $9 \ge 4q \Rightarrow 2.25 \ge q$.
The possible values for $q$ are $1, 2$. So, the pairs are $(3, 1), (3, 2)$. (2 pairs)
- Case 4: If $p = 4$, then $p^2 = 16$. The condition is $16 \ge 4q \Rightarrow 4 \ge q$.
The possible values for $q$ are $1, 2, 3, 4$. So, the pairs are $(4, 1), (4, 2), (4, 3), (4, 4)$. (4 pairs)
Total number of favorable outcomes $= 0 + 1 + 2 + 4 = 7$.
The required probability $= \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{7}{16}$.
Step 4: Final Answer:
The probability is $\frac{7}{16}$.