Contents
 • one choice • many choices: permutations • many choices: combinations • many choices: combinations • independent events • dependent events • permutations without repettitions • permutations with identical elements • combinations • coditional probabilities • and or events • two outcomes • total probability theorem • Bayes rule • union sets and probability • fundamental counting principle

 Other exercices

 Combinatorics - Probability
sample space

sample space

events

Probability originates from gambling. We talk about probability when we deal with random experiment expecting some results or events that have some certain chance to occur. Mathematics, along with set theory is the relevant tool to solve related problems. A random experiment has three properties:
1. We cannot predict with certainty the result of an experiment.
2. we can describe, before the experiment all the possible results that could happen.
3. We can repeat the same experiment as long as we want.

The set of the possibles results for a random experiment is called the sample space, and denoted by "S". There is not only one S for an experiment but many. We have to choose the complete and precise one.

For example, let's have the set :
books = {Physics1, Physics2, Physics3, Math1, Math2}.
If we consider the radom experiment "Get two books" from the set books, the set "books" can be the sample space S, but the better one, with the better description is S = books x books = {(Physics1, Physics2), (Physics1, Physics3), (Physics1, Math1), (Physics1, Math2), (Physics2, Physics3), (Physics2, Math1), (Physics2, Math2), (Physics3, Math1), (Physics3, Math2) , (Math1, Math2)}.

If with this random experiement, we are interested in "Get one Physics book and one Math book", we can write all the related paires from S in the set A = {(Physics1, Math1), (Physics1, Math2), (Physics2, Math1), (Physics2, Math2), (Physics3, Math1), (Physics3, Math2)}. The expression "Get one Physics book and one Math book" or its corresponding subset "A" is called event.

A sample space can be countable or continuous, finite or infinite. An event can be certain, impossible, simple if it contains only one element like A = {Math}, or compound if it contains more than one element. Note that we use often Venn diagrams to beter understand the relationships between events.

Random experiment
sample space: S
event
set theory
Venn diagram

 Today: : ____________

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