Question

Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the probability that :
(i) same number appears on both the dice?
(ii) different number appears on both the dice?

Hint:

The probability of an event and its complement (not the event) always sum to 1. Since (ii) is the complement of (i), \(1 - 1/6 = 5/6\).

Answer 1

Step 1: Understanding the Concept:
When two dice are thrown, the sample space consists of all possible pairs of numbers from 1 to 6. Each die has 6 faces, so the total outcomes are calculated by multiplication.
Step 2: Key Formula or Approach:
1. Total outcomes = \(6 \times 6 = 36\).
2. \(P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}\).
Step 3: Detailed Explanation:
1. Sample Space: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
2. (i) Same number (Doublets): Favorable outcomes: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} = 6. \[ P(\text{same}) = \frac{6}{36} = \frac{1}{6} \] 3. (ii) Different numbers: Favorable outcomes = Total - Same numbers = \(36 - 6 = 30\). \[ P(\text{different}) = \frac{30}{36} = \frac{5}{6} \]
Step 4: Final Answer:
The probability of same numbers is 1/6, and different numbers is 5/6.