Probability 01

Sample space and events

What entities are we operating with?

• Experiments - some kinds of actions that lead to some results
• Outcomes - possible results of experiments

Sample space (or probability space) definition

A set of all possible outcomes of our experiment can be one of three types, by cardinality

• Finite
• Countable (with cardinality )
• Uncountable (with cardinality or larger)

Examples of experiments and sample spaces

• Determining the sex of a newborn baby,
• A race of 7 horses, numbered 1 through 7,
• Flipping two coins with heads() and tails() only,
• Tossing two d6 dice,
• Measuring the lifetime of a transistor,

Event definition

Any (for uncountable case only “good”) subset If our outcome belongs to , we say that event has occurred.

Examples of event in previous experiments

• Horse number 3 won the race,
• First coin was a head,
• Sum of dice was 7,
• Transistor worked for some time, less than 5 hours,

Operations on events

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If we can perform an infinite number of experiments, we might want to define probability as This definition is very practical, but for theoretical applications it is problematic So how do we define the probability of some event?

Probability of an event definition

Properties of probability lemma

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Symmetrical, equally likely outcomes

Symmetrical outcomes are equally likely

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