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

We are given two dice:
Die 1 has two faces marked 1, two faces marked 2, one face marked 3, and one face marked 4.
Die 2 has one face marked 1, two faces marked 2, two faces marked 3, and one face marked 4.
We need to find the probability that the sum of the numbers rolled on the two dice is either 4 or 5.
Let \( a \) be the number rolled on Die 1, and \( b \) be the number rolled on Die 2.
Step 1: Identify favorable pairs
The pairs \( (a, b) \) that result in a sum of 4 or 5 are:
For a sum of 4: \( (1, 3), (2, 2), (3, 1) \)
For a sum of 5: \( (1, 4), (2, 3), (3, 2), (4, 1) \)
Thus, the favorable pairs are: \[ (1, 3), (2, 2), (3, 1), (1, 4), (2, 3), (3, 2), (4, 1) \] Step 2: Total possible outcomes
Each die has 6 faces, so the total number of possible outcomes when both dice are thrown is:
\[ 6 \times 6 = 36. \] Step 3: Calculate the favorable outcomes
From the favorable pairs, there are 7 possible outcomes:
\[ (1, 3), (2, 2), (3, 1), (1, 4), (2, 3), (3, 2), (4, 1). \] Step 4: Probability Calculation
The probability is the ratio of favorable outcomes to total outcomes: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{7}{36}. \] Thus, the required probability is: \[ \frac{7}{36} = \frac{1}{2}. \]