Question

One die has two faces marked 1, two faces marked 2, one face marked 3, and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3, and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 when both the dice are thrown together is:

• A. \( \frac{1}{2} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{4}{9} \)

Hint:

When calculating probabilities for dice rolls, list all favorable outcomes and divide by the total number of outcomes (in this case, 36).

Answer 1

The Correct Option is A

To determine the probability of obtaining a sum of 4 or 5 when rolling two dice, we first define the face distribution of each die:

• Die 1: Two faces show '1', two faces show '2', one face shows '3', and one face shows '4'.
• Die 2: One face shows '1', two faces show '2', two faces show '3', and one face shows '4'.

The total number of possible outcomes when rolling both dice, each with 6 faces, is calculated as:

\(6 \times 6 = 36\)

Next, we enumerate the favorable outcomes for each target sum.

Outcomes for a Sum of 4:

• (1,3) - Die 1 shows '1', Die 2 shows '3'.
• (2,2) - Die 1 shows '2', Die 2 shows '2'.
• (3,1) - Die 1 shows '3', Die 2 shows '1'.

The probability for each of these combinations is:

• Probability of (1,3): \(\frac{2}{6} \times \frac{2}{6} = \frac{4}{36}\)
• Probability of (2,2): \(\frac{2}{6} \times \frac{2}{6} = \frac{4}{36}\)
• Probability of (3,1): \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\)

Outcomes for a Sum of 5:

• (1,4) - Die 1 shows '1', Die 2 shows '4'.
• (2,3) - Die 1 shows '2', Die 2 shows '3'.
• (3,2) - Die 1 shows '3', Die 2 shows '2'.
• (4,1) - Die 1 shows '4', Die 2 shows '1'.

The probability for each of these combinations is:

• Probability of (1,4): \(\frac{2}{6} \times \frac{1}{6} = \frac{2}{36}\)
• Probability of (2,3): \(\frac{2}{6} \times \frac{2}{6} = \frac{4}{36}\)
• Probability of (3,2): \(\frac{1}{6} \times \frac{2}{6} = \frac{2}{36}\)
• Probability of (4,1): \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\)

Total Probability Calculation:

The combined probability of achieving a sum of 4 or 5 is the sum of the probabilities of all these favorable outcomes:

\(\left(\frac{4}{36} + \frac{4}{36} + \frac{1}{36}\right) + \left(\frac{2}{36} + \frac{4}{36} + \frac{2}{36} + \frac{1}{36}\right) = \frac{18}{36} = \frac{1}{2}\)

Therefore, the probability of obtaining a sum of 4 or 5 is \(\frac{1}{2}\). The correct option is:

• \(\frac{1}{2}\)