A probability calculation needs a description of what can happen and how those possibilities are weighted. In these questions, fairness and independence are explicit conditions of the model. Read them before counting. A fair six-sided die has six equally likely numbered results; the coin questions specify equally likely heads and tails.
Distinguish a possible outcome from a verbal category that combines outcomes. Two distinguishable coins allow heads on the first and tails on the second to be recorded separately from the reversed result. Writing the possibilities down gives you a reference for both the total count and the event being asked about.
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Keep the outcome units consistent
For a fair die numbered 1 through 6, the even results are 2, 4 and 6. Three of the six equally likely results are favourable, giving 3/6 = 1/2. Both counts refer to individual numbered outcomes. Grouping the faces into even and odd is useful only after accounting for the faces in each group.
For two distinguishable coins, record HH, HT, TH and TT, with the first letter always referring to the same coin. The independence and fairness conditions make these four ordered outcomes equally likely. HT and TH are separate outcomes even though each contains one head.
Translate at least one into a precise event
For two independent tosses of a fair coin, at least one head includes HH, HT and TH. It excludes TT, so three of the four equally likely outcomes qualify. The probability is 3/4. You can also subtract the excluded probability, 1/4, from one.
Exactly one head would exclude HH as well as TT. Write which outcomes qualify before doing the fraction calculation. This keeps the event wording separate from the arithmetic and makes it possible to check a complementary calculation against the direct list.
TRY & READ
Check your understanding with examples
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Example 1 · Probability and statistics
What is the probability of an even result when rolling one fair six-sided die numbered 1 to 6?
1/6
2/3
1/3
1/2
Read the answer and explanation
Answer: 1/2
The six outcomes are equally likely. Three, namely 2, 4 and 6, are even, so the probability is 3/6 = 1/2.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Example 2 · Probability and statistics
Two distinguishable coins are tossed independently, each with equally likely heads and tails. How many outcomes are possible?
2
3
8
4
Read the answer and explanation
Answer: 4
The ordered outcomes are HH, HT, TH and TT, giving four possibilities. HT and TH are distinct because the coins are distinguishable.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Example 3 · Probability and statistics
A fair coin is tossed independently twice. What is the probability of at least one head?
1/2
1/4
1
3/4
Read the answer and explanation
Answer: 3/4
The four equally likely outcomes are HH, HT, TH and TT. Only TT has no head, so the probability is 1 − 1/4 = 3/4.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Topics: Probability and statistics. Range: Everyday knowledge, Broader knowledge. Difficulty: Basic, Standard. These are selected initially. You can change these on the setup screen.
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