Probability · Lesson 1 of 8
Events as Sets
“Translate a question about a random experiment into a precise set of outcomes.”
• Distinguish an experiment, an outcome, a sample space, and an event. • Decide whether an event occurs by checking membership of the observed outcome. • Translate exactly, at least, and at most into sets of outcomes. • Identify impossible, sure, simple, and compound events.
From an experiment to a question
Suppose you toss a coin twice. Before the tosses, you do not know the result of either toss, but you can describe every possible result. Afterwards, you might ask whether there was exactly one head, whether the second toss was a tail, or whether both tosses matched. The experiment is the same in every case; the question changes.
An experiment whose particular result is uncertain is called a random experiment. A single result is an outcome or sample point. The collection of all possible outcomes is the sample space, usually written S. Here we record the first toss before the second, so HT means a head followed by a tail, while TH means a tail followed by a head.
The order matters because HT and TH answer different questions about the first and second tosses. Both give one head, but they are different elementary results. If you kept only the number of heads, you would record 0, 1, or 2 instead. That is a different way of describing the experiment, and those three values need not have equal chances. For now, keep the four ordered outcomes visible.
To answer “Did exactly one head appear?”, collect the outcomes that make the answer yes. They are HT and TH. The set E={HT, TH} captures the whole question: it tells us in advance which observations will count as successes for that event.
Any subset of the sample space of a random experiment. Its elements are precisely the outcomes that satisfy the condition being considered.
A subset contains only elements already in its parent set. Therefore an event cannot introduce an outcome that the experiment cannot produce. Because S is the collection of all allowed outcomes, it acts as the universal set for event descriptions in that experiment.
Reading everyday descriptions accurately
Words such as exactly, at least, and at most control which outcomes enter an event. Exactly fixes a number; at least gives a lower bound; at most gives an upper bound. A careful event description converts these verbal boundaries into membership decisions for every point in S.
| Question for two tosses | Event as a subset of S |
|---|---|
| Exactly two tails | {TT} |
| At least one tail | {HT, TH, TT} |
| At most one head | {HT, TH, TT} |
| The second toss is not a head | {HT, TT} |
| At most two tails | {HH, HT, TH, TT}=S |
| More than two tails | ∅ |
Notice that “at least one tail” and “at most one head” describe the same subset. With two tosses, having one or two tails leaves room for at most one head. Different sentences can therefore name the same event. Conversely, changing a small word can change the set: “exactly one tail” excludes TT, while “at least one tail” includes it.
Problem
For three tosses, write the events E: exactly one head, F: at least one head, and G: at most one head.
- 1.List the eight ordered outcomes: S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}. Each position records a toss.
- 2.Exactly one head can be in the first, second, or third position: E={HTT,THT,TTH}.
- 3.At least one head includes every outcome except TTT: F={HHH,HHT,HTH,THH,HTT,THT,TTH}.
- 4.At most one head allows zero or one head: G={TTT,HTT,THT,TTH}. In particular, “at most one” does not mean “exactly one”.
Do not merge HT and TH just because both contain one head. An outcome records everything specified by the experiment. An event groups the outcomes that answer the chosen question.
When does an event occur?
An event is specified before a result is observed. Once the experiment produces one outcome, you compare that result with the set. Several events can contain the same outcome, so several statements about the observation may be true at the same time.
An event E occurs exactly when the observed outcome ω belongs to E. If ω is outside E, the event does not occur.
Problem
A die is rolled and shows 3. Decide whether E: less than 4, F: even, and G: prime occur.
- 1.The sample space is S={1,2,3,4,5,6}. The observed outcome is ω=3.
- 2.E={1,2,3}; since 3∈E, E occurs.
- 3.F={2,4,6}; since 3∉F, F does not occur.
- 4.G={2,3,5}; since 3∈G, G also occurs. E and G can both occur because they share the outcome 3.
The region represents a set rather than a measured physical area. A larger drawn region does not yet tell us a probability. In a probability model, the weights attached to its outcomes determine its chance of occurring.
Events with no, one, or many outcomes
Classifying an event by its elements helps us recognise extreme and elementary cases. These labels describe the set itself. They do not depend on how the event is drawn or on how strongly someone expects it to happen.
The empty set ∅, containing no possible outcome of the experiment.
The complete sample space S, containing every possible outcome.
Problem
For one roll, classify “a multiple of 7” and “an odd or even number”.
- 1.The only allowed values are 1 through 6. None is a multiple of 7, so the first event is ∅ and is impossible.
- 2.Every value from 1 to 6 is either odd or even, so the second event is {1,2,3,4,5,6}=S and is sure.
- 3.Impossible means no sample point satisfies the condition. Sure means every sample point satisfies it. These are set statements, not guesses based on previous rolls.
An event containing exactly one sample point.
An event containing more than one sample point.
For two ordered tosses, the four simple events are {HH}, {HT}, {TH}, and {TT}. Each is a one-element set. HH is an outcome; {HH} is the simple event containing that outcome. In a finite sample space with n distinct outcomes, there are n simple events, one for each outcome.
The event {HT,TH} is compound because it contains two sample points. The three events in the earlier three-toss example are also compound: their sizes are 3, 7, and 4. A sure event may be compound when the sample space has several points. These categories are therefore not all mutually exclusive labels: “sure” describes coverage of S, while “compound” describes the number of elements.
Problem
A die is rolled. Classify A: exactly 5 and B: a number greater than 4.
- 1.A={5}, so it is a simple event.
- 2.B={5,6}, so it is a compound event.
- 3.If the die shows 5, both occur. The number of outcomes in an event is different from the number actually observed in one roll, which is always one.
You can build an event by scanning S and asking the same yes-or-no question of each outcome. This is a reliable way to check a verbal description, especially when bounds such as “at least” and “at most” are involved.
Quiz
For two ordered tosses, which set means exactly one head?
A die shows 4. Which event has occurred?
For three tosses, how many outcomes belong to “at most one head”?
Which is a simple event for a single die roll?
For two tosses, “at most two tails” is which event?
Which statement correctly distinguishes an outcome from an event?
Practice Problems
- Write the sample space for three ordered coin tosses. Form the events “exactly two heads”, “at least two heads”, and “no head”. Classify each by its number of elements.
- For two tosses, compare “at most one tail” with “at least one head”. Explain why the event sets agree.
- A die is rolled. Write the events “less than 5”, “not less than 5”, “a multiple of 8”, and “a number from 1 to 6”. Identify simple, compound, impossible, or sure cases where applicable.
- For S={red,blue,green}, give a simple event, a compound event, the impossible event, and the sure event. If blue is observed, state which of your events occur.
- For two labelled dice, explain why (2,5) and (5,2) are different outcomes even though they have the same sum.
- An event E={HH,TT} is specified for two tosses. Which observations make E occur? Give a verbal description of E and explain why “exactly one head” describes a different event.
Key Takeaways
• An outcome is one result; the sample space contains every allowed result. • An event is a subset of S and occurs when the observed outcome belongs to it. • Exactly, at least, and at most must be translated with their correct boundaries. • ∅ is impossible and S is sure; simple events have one element and compound events have more than one.
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Combining and Classifying Events