Probability theory is the branch of mathematics concerned with analysis of random phenomena. (Encyclopedia Britannica) An experiment: is any action, process or phenomenon whose outcome is subject to uncertainty (in a very loose sense!) An outcome: is a result of an experiment. Each trial of an experiment results in only one outcome! A trial: is a run of an experiment

Def : A sample space: is the set of all possible outcomes, S, of an experiment. Again, we will observe only one of these in one trial. Def : An event: is a subset of the sample space. An event occurs when one of the outcomes that belong to it occurs. Def : An elementary (simple) event: is a subset of the sample space that has only one outcome.

Sample Space and Events Examples: 5)Studying the chance of observing the faces of two dice when rolled: Experiment: Set of possible outcomes (sample space), S: Rolling two dice Goal: Observe faces of two dice die1/die (1,1)(1,2)(1,3)(1,4)(1,5)(1,6) 2(2,1)(2,2)(2,3)(2,4)(2,5)(2,6) 3(3,1)(3,2)(3,3)(3,4)(3,5)(3,6) 4(4,1)(4,2)(4,3)(4,4)(4,5)(4,6) 5(5,1)(5,2)(5,3)(5,4)(5,5)(5,6) 6(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

Some possible events (Possible subsets of S): Die 1 will have face with number 2: {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)} Sample Space and Events Examples: 5)Studying the chance of observing the faces of two dice when rolled: Die 2 will have face with number 3: {(1,3), (2,3), (3,3), (4,3), (5,3), (6,3)} An elementary (simple) event: Die 1 and Die 2 will have faces with numbers 3 and 4 respectively: {(3,4)}

Sample Space and Events Examples: 6)Studying the chance of observing the sum of faces of two dice when rolled: Experiment: Set of possible outcomes (sample space), S: Rolling two dice Goal: Observe sum of faces of two dice {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Some possible events (Possible subsets of S): Sum is less than 5: {2, 3, 4} Sum is between 3 and 9: {4, 5, 6, 7, 8} All of the above examples had a finite sample spaces; i.e. we could count the possible outcomes.

Sample Space and Events Section 2.1 Examples: 7)Flipping a coin until the first head shows up: Experiment: Set of possible outcomes (sample space), S: Flipping a coin multiple times and stop when head is observed. Goal: Observe T’s until first H {H, TH, TTH, TTTH, TTTTH, …}

Some possible events (Possible subsets of S): Sample Space and Events Examples: 7)Flipping a coin until the first head shows up: Observing at least two tails before we stop: {TTH, TTTH, TTTTH, …} Observing at most 4 tails before we stop: {H, TH, TTH, TTTH, TTTTH} Observing exactly 2 tails before we stop: {TTH}

Sample Space and Events Examples: 8)Flipping a coin until the first head shows up: Experiment: Set of possible outcomes (sample space), S: Flipping a coin multiple times and stop when head is observed. Goal: Observe number of flips needed to stop. {1, 2, 3, 4, 5, …}

Sample Space and Events Examples: 8)Flipping a coin until the first head shows up: The above two examples had a countably infinite sample spaces; i.e. we could match the possible outcomes to the integer line. Some possible events (Possible subsets of S): Observing at least two tails before we stop: {3, 4, 5, 6, …} Observing at most 4 tails before we stop: {1, 2, 3, 4, 5} Observing exactly 2 tails before we stop: {3}

Sample Space and Events Def : A discrete sample space: is a sample space that is either finite or countably infinite

Sample Space and Events Examples: 9)Flipping a coin until the first head shows up: Experiment: Set of possible outcomes (sample space), S: Flipping a coin multiple times and stop when head is observed. Goal: Observe the time t until we stop in minutes.

The above example has a continuous sample spaces; i.e. the possible outcomes belong to the real, number line. Sample Space and Events Examples: 9)Flipping a coin until the first head shows up: Some possible events (Possible subsets of S): t < 10: [0, 10) : (5, 20]

Some relations from set theory and Venn diagrams: A Set: is a collection of objects. We say x belongs to the set (event) A, if x is an element of that set; in math We also say x does not belong to the set (event) A, if x is not an element of that set; in math Event A is a subset of event B,, if every element in A is also found in B. B S A is the empty set, the null event is the sample space, the sure event

Operations to create new sets: 1)The complement of an event A, referred to as A’, is the set of all outcomes in S that do not belong to A. A’ A S The complement of

2)The union of two events A and B denoted by is the event consisting of all outcomes that are either in A, in B or in both. A S B Operations to create new sets:

3)The intersection of two events A and B denoted by is the event consisting of all outcomes that are in both A and B. A S B Operations to create new sets:

Def 1.2.5: Two events A and B are called disjoint or mutually exclusive if = A S B Also, any two elementary events are going to be disjoint!

4)Set difference: A – B, is equivalent to. A S B Operations to create new sets:

1)De Morgan’s Laws: Other important rules.