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Glacius
ProbabilityConcept reference

Sample spaces

A sample space is the set of possible outcomes for a specified experiment.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. A sample space lists the possible records of one experiment
  3. A count is a different record
  4. Two ordered choices can forbid repeats
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01A sample space lists the possible records of one experiment#

A sample space lists the possible records of one experiment. Decide what each record must preserve before listing it.

Record two A/B positions in order. Starting with A gives AA or AB; starting with B gives BA or BB.

First and second positions each allow A or B. The four distinct ordered outcomes are AA, AB, BA and BB; AB and BA retain opposite orders. Box size does not encode likelihood.First and second positions each allow A or B. The four distinct ordered outcomes are AA, AB, BA and BB; AB and BA retain opposite orders. Box size does not encode likelihood.
Figure 1First and second positions each allow A or B. The four distinct ordered outcomes are AA, AB, BA and BB; AB and BA retain opposite orders. Box size does not encode likelihood.
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Check your reasoning

First: A, B. Second: 0, 1. Repeats allowed. Choose the full ordered sample space.

  1. A{A0, A1, B0, B1}
  2. B{A0, A1, B0}
  3. C{0, 1, A, B}
Show answer and explanation
{A0, A1, B0, B1}

Keep {A0, A1, B0, B1}.

02A count is a different record#

A count is a different record. For two A/B positions, the number of A symbols can be 0, 1 or 2. The count 1 merges AB with BA, so it cannot tell you the first symbol.

Check your reasoning

Two positions, each S or F; store the count of S. Which pair hides the first symbol?

  1. ASS and FF
  2. BSS and SF
  3. CSF and FS
Show answer and explanation
SF and FS

Same count; first differs.

03Two ordered choices can forbid repeats#

Two ordered choices can forbid repeats. With L and R, only LR and RL then remain. Naming outcomes does not make them equally likely; their chances require a separate assignment.

Check your reasoning

First: U, D. Second: U, D. Repeats forbidden. Missing DU: repair the space.

  1. A{UD}
  2. B{UD, DU}
  3. C{UD, DU, UU}
Show answer and explanation
{UD, DU}

Keep UD, DU; no repeats.

Key takeaway

Define what one outcome records, then include every permitted result. Preserve any distinction the question needs.

Sources & further reading

  1. [1]
    Pishro-Nik, Introduction to Probability, 1.3.1 Random Experiments ↗Pishro-Nik, Introduction to Probability · Book

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Sample spaces.” Math behind ML. /learn/p-outcomes