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 together, each die has 6 possible outcomes.
Using multiplication principle:
Total outcomes = 6 × 6 = 36

Each outcome is written as an ordered pair (a, b), where:
a = number on first die
b = number on second die

Step 2: Writing the 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)

Total outcomes = 36

Probability formula:
P(E) = Favorable outcomes / Total outcomes

Step 3: (i) Probability of Getting Same Numbers (Doublets):
Favorable outcomes are:
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

Number of favorable outcomes = 6

P(same numbers) = 6 / 36
= 1 / 6

Step 4: (ii) Probability of Getting Different Numbers:
Total outcomes = 36
Outcomes with same numbers = 6

Favorable outcomes (different numbers) = 36 − 6
= 30

P(different numbers) = 30 / 36
= 5 / 6

Final Answer:
Probability of same numbers = 1/6
Probability of different numbers = 5/6