Question

Two dice are numbered 1,2,3,4,5,6 and 1,1,2,2,3,3 respectively. They are thrown and the sum of the numbers on them is noted. The probability of getting a sum of 5 will be :

• A. \(\frac{1}{9}\)
• B. \(\frac{1}{18}\)
• C. \(\frac{1}{6}\)
• D. \(\frac{1}{36}\)

Hint:

Die 1 (D1): \{1, 2, 3, 4, 5, 6\} Die 2 (D2): \{1a, 1b, 2a, 2b, 3a, 3b\} (labeling the repeated faces for clarity) Total outcomes = \(6 \times 6 = 36\). Pairs (D1, D2) that sum to 5:
D1=2, D2=3: (2, 3a), (2, 3b) - 2 outcomes
D1=3, D2=2: (3, 2a), (3, 2b) - 2 outcomes
D1=4, D2=1: (4, 1a), (4, 1b) - 2 outcomes Total favorable outcomes = \(2+2+2 = 6\). Probability = \(6/36 = 1/6\).

Answer 1

The Correct Option is C

Concept: Probability = (Number of favorable outcomes) / (Total number of possible outcomes). The dice are not standard identical dice; their numberings are different. Step 1: List all possible outcomes and their total number Die 1 (D1) has faces: \{1, 2, 3, 4, 5, 6\} Die 2 (D2) has faces: \{1, 1, 2, 2, 3, 3\} When two dice are thrown, the total number of possible outcomes is \(6 \times 6 = 36\). Each outcome is equally likely if we consider each face of each die distinct (e.g., D2 has two '1' faces, say 1a and 1b). Step 2: Identify favorable outcomes for getting a sum of 5 We need to find pairs (Outcome on D1, Outcome on D2) such that their sum is 5. Let's list them:
If D1 = 1, D2 needs to be 4. D2 does not have 4. (No outcome)
If D1 = 2, D2 needs to be 3. D2 has two '3' faces. So, (2, 3) and (2, 3). (2 outcomes)
If D1 = 3, D2 needs to be 2. D2 has two '2' faces. So, (3, 2) and (3, 2). (2 outcomes)
If D1 = 4, D2 needs to be 1. D2 has two '1' faces. So, (4, 1) and (4, 1). (2 outcomes)
If D1 = 5, D2 needs to be 0. D2 does not have 0. (No outcome)
If D1 = 6, D2 needs to be -1. D2 does not have -1. (No outcome) The favorable pairs (D1, D2 value) are: (2, 3) - happens in 2 ways because D2 has two '3's. (3, 2) - happens in 2 ways because D2 has two '2's. (4, 1) - happens in 2 ways because D2 has two '1's. Total number of favorable outcomes = \(2 + 2 + 2 = 6\). Step 3: Calculate the probability Total number of possible outcomes = 36. Number of favorable outcomes = 6. Probability (sum = 5) = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\) \[ P(\text{sum}=5) = \frac{6}{36} = \frac{1}{6} \]