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 ?

Q: 31. 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 ?

Total outcomes = 36 ($6 \times 6$ ordered pairs, since the dice are distinguishable by colour). - **(i) Same number on both dice**: $P = 6/36 = 1/6$, from the 6 doublets $(1,1), (2,2), \ldots, (6,6)$. - **(ii) Different numbers**: $P = 30/36 = 5/6$, using the complement rule $1 - 1/6 = 5/6$, since 'different' is the exact opposite of 'same'.

Probability

Probability with two dice

Official Answer

Total outcomes = 36 ( $6 \times 6$ ordered pairs, since the dice are distinguishable by colour).

(i) Same number on both dice: $P = 6/36 = 1/6$ , from the 6 doublets $(1,1), (2,2), \ldots, (6,6)$ .
(ii) Different numbers: $P = 30/36 = 5/6$ , using the complement rule $1 - 1/6 = 5/6$ , since 'different' is the exact opposite of 'same'.

sample space36 outcomestwo dicedoubletsP(same) = 1/6P(different) = 5/6complement ruleequally likely

Marking Scheme

11 mark: Writing/indicating the sample space and stating the total number of outcomes = 36.
21 mark: (i) Identifying the 6 doublets and computing $P(\text{same number}) = 6/36 = 1/6$ .
31 mark: (ii) Computing $P(\text{different numbers}) = 30/36 = 5/6$ (or $1 - 1/6 = 5/6$ ). Both simplified fractions required for full marks.

Hint

There are $6 \times 6 = 36$ equally likely outcomes. Count the 6 doublets for part (i); for part (ii) use $1 - P(\text{same})$ .

Quick Oral Answer

Two different-coloured dice give 36 equally likely ordered outcomes. Six of them are doublets, so the probability of the same number is 6 by 36, that is one-sixth; different numbers is the rest, 30 by 36 or five-sixths.

Analysis & Explanation

Tests the foundation of theoretical probability: correctly building the sample space and counting favourable outcomes.

Concept & key points

'Two dice of different colours' signals the dice are distinguishable, so $(2,3)$ and $(3,2)$ are separate outcomes — total outcomes = $6 \times 6 = 36$ , not 21.
'Same number' (doublets) and 'different numbers' are complementary events, so their probabilities must add to 1 — a useful in-exam check.

Common mistakes

Using 21 as the total by treating the dice as indistinguishable, instead of the correct 36 ordered outcomes.
Painstakingly listing all 30 'different number' outcomes instead of using the faster complement rule $1 - P(\text{same})$ .

Real-world relevance

Two-dice sample-space reasoning is the classic gateway to independent events, and underlies board-game probability, risk assessment, and Monte-Carlo simulation in computing.

Common Mistakes

1Taking the total number of outcomes as 21 by treating the two dice as indistinguishable — the correct equally-likely total is 36.
2Forgetting to simplify: leaving the answer as $6/36$ and $30/36$ instead of $1/6$ and $5/6$ .
3Miscounting the 'different' outcomes by trying to list all 30 instead of using the faster complement $1 - 1/6$ .

Interesting Facts

When two dice are rolled, the sum 7 is the single most likely total (probability $6/36 = 1/6$ ) because it can be formed in the most ways — a fact that shapes strategy in the game of Craps.

The probability of a 'doublet' (same number on both dice) is $1/6$ — exactly the same as rolling any one chosen number on a single die.

Systematic study of dice probabilities began with Gerolamo Cardano in the 16th century and was formalised by Blaise Pascal and Pierre de Fermat in 1654, founding modern probability theory.

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Frequently Asked Questions

Why are there 36 outcomes and not 21?

Because the dice are distinguishable (different colours), the pair $(2,3)$ is different from $(3,2)$ . Each of the 6 faces of the first die can pair with each of the 6 faces of the second, giving $6 \times 6 = 36$ equally likely ordered outcomes. 36 is the correct total for probability calculations.

What are 'doublets'?

Doublets are outcomes where both dice show the same number: $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$ — six outcomes in all. Their probability is $6/36 = 1/6$ .

How do the two probabilities relate?

'Same number' and 'different numbers' are complementary events — every outcome is one or the other. So their probabilities add to 1: $1/6 + 5/6 = 1$ . This is a quick way to check your answer and to find P(different) as $1 - P(\text{same})$ .