Basic Concepts of Probability

Experiments, Sample Spaces, and Events

Experiment: A process or action with uncertain outcomes that can be repeated under identical conditions.

Example: Tossing a coin, rolling a die, or drawing a card from a deck.

Sample Space (): The set of all possible outcomes of an experiment.

For tossing a coin:

For rolling a die:

Event (): A subset of the sample space, representing outcomes of interest.

Rolling an even number on a die:

Drawing a red card from a standard deck:

Set Theory and Events

Probability theory uses set operations to describe relationships between events:

Union (): The event that either , , or both occur.

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Rolling a number less than 3 () or an even number ():

Intersection (): The event that both and occur.

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For and :

Complement (): The event that does not occur.

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If for rolling a die, then:

Mutually Exclusive Events: Events that cannot occur simultaneously.

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Rolling an odd number and rolling an even number on a die:

Axioms of Probability

Probability is based on Kolmogorov's three axioms:

Non-Negativity: for any event .

Normalization: , where is the sample space.

Additivity: For mutually exclusive events

Probability Measures

A probability measure assigns a value to each event in the sample space, satisfying the axioms of probability. Let be the sample space with probabilities . A valid probability measure satisfies:

for all .

For an event , .

Examples

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Tossing Two Coins:

Sample Space:

Event:

Probability:

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Rolling a Fair Die:

Sample Space:

Event:

Probability:

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If two dice are rolled, what is the probability that the sum is 7 given that one die shows 4?:

Sample Space:

Event : Sum = 7 and one die shows 4:

Probability: