Question

Two identical cubic dice are rolled simultaneously. Which of the following is the probability that at least one of the face values of the two dice is greater than 3?

• A. \(\dfrac{1}{2}\)
• B. \(\dfrac{3}{4}\)
• C. \(\dfrac{1}{4}\)
• D. \(\dfrac{3}{8}\)

Hint:

Use the complement rule: find the probability both dice show 3 or less, then subtract from 1.

Answer 1

The Correct Option is B

Step 1: Identify the sample space:
When two identical cubic dice are rolled simultaneously, each die can show a face value from 1 to 6. The total number of equally likely outcomes is \(6 \times 6 = 36\).
Step 2: Find the complement event:
We want the probability that at least one face value is greater than 3, meaning at least one die shows 4, 5 or 6. It is easier to work with the complement, which is the event that neither die shows a value greater than 3, that is both dice show a value from the set \(\{1, 2, 3\}\).
Step 3: Compute the probability of the complement:
For each die, the probability of getting a value less than or equal to 3 is \(\dfrac{3}{6} = \dfrac{1}{2}\). Since the two dice are rolled independently, the probability that both dice show a value less than or equal to 3 is \(\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}\).
Step 4: Apply the complement rule:
The probability that at least one die shows a value greater than 3 is 1 minus the probability of the complement event, so \(P = 1 - \dfrac{1}{4} = \dfrac{3}{4}\).
Step 5: Check the options:
Option A gives \(\dfrac{1}{2}\), which is not correct since it does not account for both dice correctly. Option B gives \(\dfrac{3}{4}\), which matches our computed value. Option C gives \(\dfrac{1}{4}\), which is actually the probability of the complement event, not the required event. Option D gives \(\dfrac{3}{8}\), which does not correspond to any valid computation for this problem.
Final Answer:
\[ \boxed{P = \dfrac{3}{4}} \]