Understanding Events: The Core of Probability Experiments
From Experiments to Outcomes to Events
Probability begins with an experiment[2], which is any process that leads to well-defined results. Think of tossing a coin, rolling a die, or drawing a card from a deck. Each possible result of an experiment is called an outcome.
An event is simply a specific collection of these outcomes that we are interested in. It is the "outcome that is being tested for." The event is the question we ask about the experiment.
$P(E) = \frac{\text{Number of favorable outcomes for } E}{\text{Total number of possible outcomes in the experiment}}$
This formula works when all outcomes are equally likely.
Consider the experiment of rolling a standard six-sided die. The possible outcomes are: $\{1, 2, 3, 4, 5, 6\}$.
- Event A: "Rolling an even number." The outcomes being tested for are $\{2, 4, 6\}$.
- Event B: "Rolling a number greater than 4." The outcomes are $\{5, 6\}$.
- Event C: "Rolling a 3." This event tests for a single outcome: $\{3\}$.
Simple Events vs. Compound Events
Not all events are the same. They are classified based on how many outcomes they contain.
A Simple Event (or elementary event) is an event that contains exactly one outcome from the sample space. It is the most basic result you can get from an experiment. In our die-rolling example, Event C ("Rolling a 3") is a simple event.
A Compound Event is an event that consists of two or more simple events. It combines multiple outcomes. Events A (even number) and B (greater than 4) are both compound events. We often form compound events using the words "or," "and," and "not."
| Event Description | Outcomes Tested For | Type of Event | Probability |
|---|---|---|---|
| Rolling a 5 | $\{5\}$ | Simple | $P=\frac{1}{6}$ |
| Rolling an odd number | $\{1, 3, 5\}$ | Compound | $P=\frac{3}{6}=\frac{1}{2}$ |
| Rolling a number less than 3 | $\{1, 2\}$ | Compound | $P=\frac{2}{6}=\frac{1}{3}$ |
| Rolling a 7 | $\{\}$ (None) | Impossible | $P=\frac{0}{6}=0$ |
Combining Events: Unions and Intersections
We can create new events by combining two or more events. This is where the words "OR" and "AND" come into play in probability.
The Union of two events A and B (written as $A \cup B$) is the event that either A or B (or both) occur. It combines all outcomes from both events.
The Intersection of two events A and B (written as $A \cap B$) is the event that both A and B occur simultaneously. It includes only the outcomes common to both events.
Example: Let's use our die-rolling events: A = {2, 4, 6} (even), B = {5, 6} (greater than 4).
- Union (A OR B): $A \cup B = \{2, 4, 5, 6\}$. The probability is $P(A \cup B) = \frac{4}{6} = \frac{2}{3}$.
- Intersection (A AND B): $A \cap B = \{6\}$. The only outcome that is both even AND greater than 4 is 6. The probability is $P(A \cap B) = \frac{1}{6}$.
Events in Action: Predicting Card Draws
Let's apply these ideas to a classic experiment: drawing one card from a standard 52-card deck. The deck has four suits (hearts, diamonds, clubs, spades) and 13 ranks (Ace through King).
Define some events:
- Event H: "Drawing a Heart." This event has 13 favorable outcomes.
- Event K: "Drawing a King." This event has 4 favorable outcomes (one from each suit).
- Event R: "Drawing a red card." This includes all Hearts and Diamonds, totaling 26 favorable outcomes.
We can calculate probabilities and combine events:
- $P(H) = \frac{13}{52} = \frac{1}{4}$
- $P(K) = \frac{4}{52} = \frac{1}{13}$
- What is $P(H \cap K)$? This is the event "Drawing the King of Hearts." There is only 1 such card, so $P = \frac{1}{52}$.
- What is $P(H \cup R)$? Since all Hearts are already red cards, the union of H and R is just event R itself (26 cards). So $P = \frac{26}{52} = \frac{1}{2}$.
Special Types of Events: Impossible and Certain
Two special events mark the extremes of probability.
The Impossible Event contains no outcomes from the sample space. Its probability is always 0. In our die example, the event "Rolling a 7" is impossible.
The Certain Event (or sure event) contains all possible outcomes from the sample space. Its probability is always 1. For a die roll, the event "Rolling a number between 1 and 6" is certain.
This gives us the fundamental range of all probabilities: for any event $E$, $0 \le P(E) \le 1$.
Important Questions
Yes. Every single outcome can be considered a simple event. Furthermore, any collection of outcomes (including the empty collection and the full collection) can be defined as an event. When solving problems, you define the event based on the question being asked.
Carefully read the description of the event. Translate the words into a specific list of results from the experiment's sample space. For "rolling a prime number on a die," you would list the prime numbers between 1 and 6: {2, 3, 5}. This list is your event.
No. Probability is always a number between 0 and 1, inclusive. A probability of 0 means the event is impossible. A probability of 1 means the event is certain. Values like 1.5 or -0.2 are meaningless in standard probability.
Footnote
[1] Event: In probability, an event is a set of outcomes of an experiment to which a probability is assigned.
[2] Experiment (Probability Experiment): A process or procedure that yields one of a set of possible outcomes. Examples include tossing a coin, rolling a die, or drawing a card.