Question

If two numbers \( p \) and \( q \) are chosen randomly from the set \( \{1, 2, 3, 4\} \) with replacement, what is the probability that \( p^2 \geq 4q \)?

• A. \( \frac{1}{4} \)
• B. \( \frac{3}{16} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{7}{16} \)

Hint:

When dealing with probability problems involving inequalities, first list the favorable outcomes and then divide by the total number of possible outcomes for an accurate result.

Answer 1

The Correct Option is D

Step 1: Determine the total number of possible outcomes. When selecting \( p \) and \( q \) from the set \( \{1, 2, 3, 4\} \) with replacement, the possible pairs are: \[ S = \{(1, 1), (1, 2), \dots, (4, 4)\}. \] Thus, the total number of outcomes is: \[ n(S) = 4 \times 4 = 16. \] Step 2: Identify the favorable outcomes where \( p^2 \geq 4q \). List the pairs satisfying \( p^2 \geq 4q \): \[ E = \{(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (4, 4)\}. \] The number of favorable outcomes is: \[ n(E) = 7. \] Step 3: Calculate the probability. The probability of the event \( E \) is given by: \[ P(E) = \frac{n(E)}{n(S)} = \frac{7}{16}. \] Final Answer: The probability is: \[ \boxed{\frac{7}{16}}. \]