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:

Using the complement rule $P(E') = 1 - P(E)$ is always faster and less prone to errors than manually counting the remaining 30 outcomes.

Answer 1

Step 1: Understanding the Question:
Two dice are rolled simultaneously. We need to:
1. List the entire sample space (all possible outcomes).
2. Calculate the probability of getting the same number on both dice (doublets).
3. Calculate the probability of getting different numbers on both dice.

Step 2: Key Formula or Approach:
The probability of an event $E$ is given by:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
The total number of outcomes when two dice are thrown is $6 \times 6 = 36$.

Step 3: Detailed Explanation:

• 1. Write down the sample space containing all 36 possible outcomes:
\[ S = \begin{Bmatrix} (1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6) \\ (2,1), & (2,2), & (2,3), & (2,4), & (2,5), & (2,6) \\ (3,1), & (3,2), & (3,3), & (3,4), & (3,5), & (3,6) \\ (4,1), & (4,2), & (4,3), & (4,4), & (4,5), & (4,6) \\ (5,1), & (5,2), & (5,3), & (5,4), & (5,5), & (5,6) \\ (6,1), & (6,2), & (6,3), & (6,4), & (6,5), & (6,6) \end{Bmatrix} \]
Total number of outcomes $n(S) = 36$.

• 2.

Evaluate Part (i): Same number appears on both dice (Doublets):
- Favorable outcomes are:
\[ E_1 = \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\} \]
- Number of favorable outcomes $n(E_1) = 6$.
- Calculate probability:
\[ P(E_1) = \frac{n(E_1)}{n(S)} = \frac{6}{36} = \frac{1}{6} \]

• 3.

Evaluate Part (ii): Different number appears on both dice:
- This event is the complement of the event in Part (i).
- Therefore, we can use the complement rule:
\[ P(E_2) = 1 - P(E_1) \]
- Substitute the value:
\[ P(E_2) = 1 - \frac{1}{6} = \frac{5}{6} \]
- Alternatively, the number of outcomes with different numbers is $36 - 6 = 30$.
- Probability:
\[ P(E_2) = \frac{30}{36} = \frac{5}{6} \]


Step 4: Final Answer:
(i) The probability that the same number appears on both dice is $\frac{1}{6}$.
(ii) The probability that a different number appears on both dice is $\frac{5}{6}$.