Probability — Part 1: Sample spaces and events: visual lesson, CBSE Class 11 Mathematics

Lesson

Random experiments and their outcomes

Before we can measure chance we must say exactly what can happen. This part builds the language: outcomes, sample spaces and events as sets.

In this lesson you will

Describe a random experiment and write its sample space (formative)
Write events as subsets and combine them with or, and, not and but not
Decide whether two events are mutually exclusive or exhaustive
Split a sample space into mutually exclusive and exhaustive events

The sample space lists every outcome once; its size is $n(S)$.
Random experiment	Sample space $S$	$n(S)$
Toss a coin once	$\{\text{H}, \text{T}\}$	2
Toss a coin; on a head throw a die, on a tail toss the coin again	$\{\text{H1}, \text{H2}, \ldots, \text{H6}, \text{TH}, \text{TT}\}$	8
Draw two of the tickets 1, 2, 3, 4 one after the other, without putting the first back	$\{(1, 2), (1, 3), (1, 4), (2, 1), \ldots, (4, 3)\}$	12
Pick a pair of people from A, B, C, D (order does not matter)	$\{\text{AB}, \text{AC}, \text{AD}, \text{BC}, \text{BD}, \text{CD}\}$	6

Concept

Counting the outcomes

Before counting, decide two things: does the order matter, and is an item put back before the next draw? The answer changes the size of the sample space.

Decide whether order matters and whether items are put back before you count $n(S)$.
Way of choosing 2 from 4 items	Example outcome	Number of outcomes
One after the other, item put back (order matters)	$(a, a)$	16
One after the other, item not put back (order matters)	$(a, b)$	12
Both together (order does not matter)	$\{a, b\}$	6

Worked example

Worked example 1: a two-stage experiment

A coin is tossed. If it shows a head, a die is thrown; if it shows a tail, the coin is tossed again. Write the sample space.

Head first — $\text{H1}, \ldots, \text{H6}$
Tail first — $\text{TH}, \text{TT}$
Size — $n(S) = 6 + 2 = 8$

The outcomes are not equally likely, but the sample space is still the list of all of them.
Step	Working
Head first	$\text{H1}, \text{H2}, \text{H3}, \text{H4}, \text{H5}, \text{H6}$ (a die is then thrown)
Tail first	$\text{TH}, \text{TT}$ (the coin is tossed again)
Sample space	$S = \{\text{H1}, \ldots, \text{H6}, \text{TH}, \text{TT}\}$
Size	$n(S) = 6 + 2 = 8$

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