Grade 10 · Core Mathematics · Lesson 4 of 12

Generating Probability Space of Events

A Grade 10 Core Mathematics CBE lesson from the sub-strand Probability 1. In it, learners define probability space, sample space, and outcome in the context of random experiments.

Strand Statistics and ProbabilitySub-strand Probability 1Lesson code 10CM-32-L440 minutes · 3 slides

KICD learning outcome (c)

Generate probability space of different events.

By the end of this lesson, learners should be able to

Define probability space, sample space, and outcome in the context of random experiments.
Systematically list all possible outcomes for combined coin tosses and multi-stage events.
Construct and interpret sample space tables for sums and products of scores from two dice or spinners.

Key vocabulary

Probability space: The set of all possible outcomes resulting from a random experiment or trial.; The probability space for flipping two coins consists of four ordered pair outcomes.
Sample space: Another term for probability space, representing all possible results of an experiment.; Writing down the sample space helps ensure no outcome is omitted in analysis.
Sample space table: A rectangular grid listing outcomes of two independent events along rows and columns.; The sample space table displays twenty-four sums for the two rolled dice.
Trial: A single performance or observation of a random experiment yielding an outcome.; Each trial of spinning the spinner and flipping the coin gives one pair.

Lesson preview

Slide 1 · Understanding Probability Spaces and Coin TossesOpening · 12 min

Figure 1: A tree diagram mapping out the possible outcomes of tossing two coins to generate an ordered pair list of the sample space.

Constructing a probability space for a random experiment means writing down every single outcome that could occur. For instance, when flipping two distinct coins simultaneously, we can denote the heads and tails of the first coin as $H_{1}$ and $T_{1}$ , and those of the second coin as $H_{2}$ and $T_{2}$ . The complete set of paired results forms the sample space, also termed the outcome space or possibility space. Use Figure 1 to see how a tree diagram represents this sample space.

A probability space or sample space contains every possible outcome of a random experiment.
Labelling coin outcomes like $H_{1}, T_{1}$ and $H_{2}, T_{2}$ prevents confusion during outcome listing.
Two tossed coins produce four combined outcomes: $(H_{1}, H_{2}), (H_{1}, T_{2}), (T_{1}, H_{2}),$ and $(T_{1}, T_{2})$ .
The total number of outcomes for independent events is found by multiplying individual outcome counts.

Worked example. Identify the possible outcomes for Coin 1 as {

H_{1}, T_{1}

} and for Coin 2 as {

H_{2}, T_{2}

}. Pair each outcome of Coin 1 with each outcome of Coin 2 systematically. List the resulting ordered pairs:

(H_{1}, H_{2}), (H_{1}, T_{2}), (T_{1}, H_{2}), (T_{1}, T_{2})

. Count the total number of distinct outcomes:

2 \times 2 = 4

. Answer: Probability space = { $(H_{1}, H_{2}), (H_{1}, T_{2}), (T_{1}, H_{2}), (T_{1}, T_{2})$ }, total = 4 outcomes.

Check question: A coin with faces $H, T$ and a 3-sided spinner with numbers $1, 2, 3$ are spun and tossed together. List all outcomes in the sample space and state the total number of outcomes.

{ $(H, 1), (H, 2), (H, 3), (T, 1), (T, 2), (T, 3)$ }; total = 6 outcomes
{ $(H), (T), (1), (2), (3)$ }; total = 5 outcomes
{ $(H, 1), (T, 2)$ }; total = 2 outcomes
{ $(H, 1), (H, 2), (H, 3)$ }; total = 3 outcomes

Show answer

{ $(H, 1), (H, 2), (H, 3), (T, 1), (T, 2), (T, 3)$ }; total = 6 outcomes. Pairing each of the 2 coin outcomes with each of the 3 spinner numbers gives $2 \times 3 = 6$ total outcomes.

Slide 2 · Constructing Sample Space Tables for DiceModel · 14 min

Figure 2: A grid table showing the sample space of sums from rolling a standard six-sided die and a special four-sided die.

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All 3 slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 11 lessons in this sub-strand

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Generating Probability Space of Events

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 12 lessons in Probability 1