Question

Two different dice are rolled together. The probability that both the obtained numbers are less than 4, is

• A. \(\frac{2}{9}\)
• B. \(\frac{7}{36}\)
• C. \(\frac{1}{4}\)
• D. \(\frac{2}{3}\)

Hint:

Since rolling two separate dice are independent events, you can find the individual probabilities and multiply them:
- Probability of getting a number less than 4 on the first die is \(\frac{3}{6} = \frac{1}{2}\).
- Probability of getting a number less than 4 on the second die is \(\frac{3}{6} = \frac{1}{2}\).
- Since they are independent, the total probability is:
\[ P = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \] This product rule is extremely fast and prevents you from having to list out all the coordinate pairs!

Answer 1

The Correct Option is C

Step 1: Set up the sample space as a grid.
Rolling two dice gives a 6 by 6 grid of 36 equally likely outcomes, one for each pair (first die, second die).
Step 2: Mark the favourable region.
The condition "both numbers less than 4" restricts the first die to \(\{1,2,3\}\) and the second die to \(\{1,2,3\}\), which is a smaller 3 by 3 block inside the full grid.
Step 3: Compare the areas of the two grids.
The favourable block has \(3 \times 3 = 9\) outcomes, while the full grid has \(6 \times 6 = 36\) outcomes.
Step 4: Compute the probability as a ratio of grid sizes.
\[ P = \frac{9}{36} = \frac{1}{4} \]
This matches option (C).
\[ \boxed{P = \frac{1}{4}} \]