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Single-step theoretical probability and complements

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Six otherwise identical cards are labelled red, blue, red, green, red and yellow. They are shuffled and one is selected. What is the probability of red?

Year 10, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Single-step theoretical probability

A theoretical model assigns probabilities to possible outcomes. Favourable over total is valid when the counted outcomes are equally likely

\[\Pr(A)=\frac{|A|}{|S|}\]
  1. Define the notation S is the sample space and A is an event. Vertical bars count their outcomes
  2. Count positions, not names Repeated labels can have different probabilities because they occupy different numbers of outcomes
  3. Complement Everything outside A is its complement \(\displaystyle \Pr(A^{\prime})=1-\Pr(A)\)
  4. Disjoint alternatives Add favourable counts for categories that cannot occur together in the same trial
  5. Check assumptions Unequal sectors or biased selection cannot be treated as equally likely outcomes without justification

Worked example

Ten identical shuffled cards carry four red, three blue and three green labels. Find the probability of red or blue, then its complement

\[\Pr(A)=\frac{|A|}{|S|}\]
  1. \(\displaystyle |S|=10\)Count all individual cards
  2. \(\displaystyle |A|=4+3=7\)Red and blue are disjoint results of one selection
  3. \(\displaystyle \Pr(A)=\frac{7}{10}\)Divide favourable outcomes by all equally likely outcomes
  4. \(\displaystyle \Pr(A^{\prime})=1-\frac{7}{10}=\frac{3}{10}\)The complement consists of the three green cards

Common mistakes

Counting labels
✕\(\displaystyle \Pr(\text{red})=1/3\)✓\(\displaystyle \Pr(\text{red})=4/10\)
In the ten-card example, red appears on four individual cards
Excluding the wrong event
✕\(\displaystyle \Pr(A^{\prime})=\Pr(A)\)✓\(\displaystyle \Pr(A^{\prime})=1-\Pr(A)\)
Equal event and complement probabilities occur only when each is one-half
Including the endpoint
✕\(\displaystyle \{x:x<4\}=\{1,2,3,4\}\)✓\(\displaystyle \{x:x<4\}=\{1,2,3\}\)
These statements use the six die values from 1 to 6. Less than is strict

Difficulty levels

introIndividual cards, outcome language and one event or its complement
standardCategory counts, disjoint alternatives and complements
challengeNumber predicates, matching outcome positions and complementary probabilities
examMixed source representations, event subsets and exact probability forms

Questions about this topic

Why count repeated labels separately?+

The randomly selected objects or positions are outcomes. A shared label does not merge distinct selectable objects

Can all colours be assumed equally likely?+

Only when the model supports that assumption. Equal objects need equal counts per colour; a fair spinner needs equal total angles

What does neither mean?+

Exclude both named categories. With one draw from disjoint categories, count the remaining objects

Is 1 prime?+

No. A prime integer is greater than 1 and has exactly two positive factors

Is an experimental estimate the same thing?+

No. Theoretical probability comes from a model; an observed relative frequency comes from completed trials and can vary

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 4: Probability

In this chapter

Describe and order probability valuesSingle-step theoretical probability and complements4A Probability scale and experiments: Experimental probability from observed frequenciesFinite events, probability assignments and complements4A Probability scale and experiments: Design and interpret a probability simulation4B Venn diagrams and sets: Venn regions, set language and probabilityCompound events, the addition rule and mutual exclusivityComplete and use a two-way probability tableConstruct and complete two-way frequency tablesConvert between Venn diagrams and two-way tablesConditional probability from exact representations4D Conditional probability: Conditional probability on Venn diagramsTwo-step sample-space tables with and without replacementRepresent a multi-stage finite sample spaceTree-diagram probability for two-stage experimentsMulti-stage probability, multiplication and total probability4G Independence and simulations: Independence on Venn diagramsTest exact independence4G Independence and simulations: Design and interpret a probability simulation4G Independence and simulations: Experimental probability from observed frequencies