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Mathematics · Probability conditions

Probability conditions: does the next draw change?

Listing outcomes is only useful after choosing the model described by the question. Removing an item can change the next draw. Two dice create ordered pairs of results. Independent coin tosses keep the next probability fixed even after a striking sequence. These examples focus on those conditions rather than repeating the first probability guide’s single-event counts.

Underline the rule governing each trial before writing a fraction. Ask what is distinguishable, which outcomes are equally likely, and whether an earlier result changes what remains. The three examples are hypothetical models with explicit rules; no observed experiment is needed to establish those rules.

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Without replacement, track the remaining collection

Label the two red items R1 and R2 and the blue item B. Uniformly choosing two without replacement gives three equally likely unordered pairs: R1R2, R1B and R2B. Two pairs have different colours, giving probability 2/3. The labels distinguish physical items even when their colours match.

If you list ordered draws instead, there are six equally likely ordered pairs and four favourable ones. That gives the same 2/3. Keep the numerator and denominator in the same representation, and do not count R1R1 as possible when replacement is excluded.

Ordered dice results and independent future tosses

Two fair dice rolled independently have 36 equally likely ordered results. The sum seven occurs in six: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). Thus its probability is 6/36 = 1/6. Different possible sums do not contain equal numbers of pairs.

For the independent fair coin, five heads in a row do not change the next toss’s probability of tails: it remains 1/2. This conclusion uses the stated independence and fairness, rather than a promise that every unknown physical coin behaves that way.

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Example 1 · Probability and statistics

Two items are sampled uniformly without replacement from a collection of two red items and one blue item. What is the probability that their colours differ?

  1. 1/3
  2. 4/9
  3. 1/2
  4. 2/3
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Answer: 2/3

There are three equally likely unordered pairs of distinct items. Two contain the blue item and one red item, so the probability is 2/3.

Sources for this example

  • English mathematics: original derivations and adaptation record

    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 fair six-sided dice are rolled independently. What is the probability that their sum is 7?

  1. 1/7
  2. 7/36
  3. 1/12
  4. 1/6
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Answer: 1/6

There are 6 × 6 = 36 equally likely ordered pairs. Six have sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). The probability is 6/36 = 1/6.

Sources for this example

  • English mathematics: original derivations and adaptation record

    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 and has shown heads five times in a row. What is the probability that the next toss is tails?

  1. 1/6
  2. 5/6
  3. 1/2
  4. 1
Read the answer and explanation

Answer: 1/2

Independence means earlier outcomes do not change the next toss's probabilities. A fair coin still has probability 1/2 of tails.

Sources for this example

  • English mathematics: original derivations and adaptation record

    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

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