Probability
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Track your progress:
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Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Probabilistic Events
Trial, Outcome and Event
In probability, a trial is any procedure with an uncertain result, such as flipping a coin, rolling a die, or drawing a card. Each possible result of a trial is called an outcome.
An event is a collection of one or more outcomes, representing scenarios we're interested in, such as rolling an even number or drawing a red card. Events are the probabilities we calculate, and are typically denoted with letters such as A, so that the "probability of an event A" is given by P(A).
Sample Space
All possible outcomes from a single trial form the sample space, denoted U.
The overall probability of the sample space, denoted P(U), is 1. This expresses the idea that if you perform a trial, something must happen.
Theoretical Probability
Theoretical probability is calculated based on reasoning or mathematical principles—it's what we expect to happen. When outcomes are equally likely, the probability of an event is given by
where n(A) is the number of outcomes in event A, and n(U) is the total number of outcomes in the sample space.
Experimental Probability
Experimental probability (or relative frequency) is found by actually conducting trials and observing outcomes. The relative frequency is calculated by:
While theoretical probability tells us what's expected, experimental probability tells us what's observed.
Complementary Event
The complement of an event A, denoted A′, is the event that A does not happen. Since A either happens or it doesn't, then exactly one of A and A′ must happen for each trial:
This expresses the idea that the probability of the entire outcome space is 1.
Combined Events
Intersection of probabilities
The intersection is the event where both A and B occur simultaneously, denoted A∩B.
Union of probabilities
The union is the event that at least one of A or B occurs. The union is denoted A∪B and has probability
This formula is sometimes referred to as the inclusion-exclusion rule. It is often rearranged in the form
Mutually exclusive events
Events are mutually exclusive if they cannot both occur at once. In this case, the intersection probability is zero:
And therefore
Conditional Probability
Conditional probability is the probability of event A happening given we already know event B has occurred. It's calculated by taking the probability that both events occur, divided by the probability of the known event B:
Notice that we can rearrange this formula to get a general formula for the probability of multiple events,
Venn Diagrams
A Venn diagram is a visual tool used to illustrate relationships between sets or events. Each event is represented by a circle, and overlaps between circles represent shared outcomes. This helps you clearly see intersections (outcomes common to events), unions (all outcomes in any event), and mutually exclusive events (no overlap).
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Venn diagrams are often filled in with numbers representing the number of samples in each category.
Dependence
Selection with & without replacement
In probability, a selection is the action of choosing one or more items from a set or group. We can perform selections either with replacement or without replacement.
In selection with replacement, each chosen item is returned to the original group before the next choice, keeping the probabilities constant across selections.
In selection without replacement, the chosen items are removed from the group, causing probabilities to change after each pick because the number of available items decreases.
This difference significantly impacts how probabilities are calculated, especially in problems involving multiple selections.
Tree Diagram
A tree diagram visually represents possible outcomes of multiple-step probabilistic events. Each branch splits to represent different outcomes of each stage, with probabilities written along the branches. Tree diagrams simplify complex probability problems, clearly showing how probabilities combine at each step.
To find the probability of a particular combination of events -- a node, or "leaf," on the far right side of the tree diagram -- multiply probabilities along its branches. To find the probability of an event with multiple outcomes, sum the probabilities of each relevant branch.
For example, a tree diagram portraying the outcomes of an experiment with two trials, where the outcomes are two complementary events, looks like:
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