Probability Theory:

1. Probability Theory: Probability theory is the branch of mathematics concerned with analysis of random phenomena. (Encyclopedia Britannica) Probability theory: “is the branch of mathematics concerned with” the study and modeling “of random phenomena.” (I added my two cents.) Agreeing with Lindley, this branch allows us to understand and quantify uncertainty. (Dennis V. Lindley. 2006. Understanding uncertainty.) Examples of uncertain phenomena.

2. Sample Space and Events An experiment: is any action, process or phenomenon whose outcome is subject to uncertainty (in a very loose sense!) A trial: is a run of an experiment An outcome: is a result of an experiment. Each trial of an experiment results in only one outcome!

3. Sample Space and Events Examples: Studying the chance of observing a head (H) or a tail (T) when flipping a coin once:

4. Sample Space and Events Some assumptions to keep in mind: Experiments are assumed to be repeatable under, essentially, the same conditions. (Not always possible and not always necessary.) Which outcome we will attain in a trial is uncertain, is not known before we run the experiment, but, The set of all possible outcomes can be specified before performing the experiment.

5. Sample Space and Events Examples: Studying the chance of observing a head (H) or a tail (T) when flipping a coin once:

6. Sample Space and Events Def. 1.2.1: 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. 1.2.3: An event: is a subset of the sample space. An event occurs when one of the outcomes that belong to it occurs. Def. 1.2.4: An elementary (simple) event: is a subset of the sample space that has only one outcome.

7. Sample Space and Events Examples: Studying the chance of observing a head (H) or a tail (T) when flipping a coin once: Observe whether the coin will be H or T Goal: Experiment: Flipping a coin Set of possible outcomes (sample space), S: {H,T} Collection of possible events (Possible subsets of S): { , {H}, {T}, {H,T}=S} : is the empty set. S: the sure event.

8. Sample Space and Events We are interested in the chance that some event will occur. Probability Population Sample

9. Sample Space and Events Before observing an outcome of an experiment (i.e. before any trials) each of the possible events will have some chance of occurring. In probability we quantify this chance and hence speculate about what an event might look like given what we might know about the experiment and the associated population.

10. Sample Space and Events Examples: Studying the chance of observing a head (H) when flipping a coin once: Observe a H Goal: Experiment: Flipping a coin Set of possible outcomes (sample space), S: {1, 0}, {H,T} or {S, F} Collection of possible events (Possible subsets of S): { , {1}, {0}, {1,0}=S}

11. Sample Space and Events Examples: Forecasting the weather for each of the next three days on the Palouse: Interested in whether you should bring an umbrella or not. Goal: Experiment: Observing whether the weather is rainy (R) or not (N) in each of the next three days on the Palouse. Set of possible outcomes (sample space), S: {NNN, RNN, NRN, NNR, RRN, RNR, NRR, RRR}

12. Sample Space and Events Examples: Forecasting the weather the next three days on the Palouse: Collection of possible events (Possible subsets of S): { , {NNN}, {RNN}, …, {NNN, RNN}, {NNN, NRN},…, {NNN, RNN, NRN},…, {NNN, RNN, NRN, NNR, RRN, RNR, NRR, RRR}=S} {{NNN}, {RNN}, {NRN}, {NNR}, {RRN}, {RNR}, {NRR}, {RRR}} are called the elemintary events. S, is called the sure event (as well as the sample space.)

13. Sample Space and Events Examples: Forecasting the weather the next three days on the Palouse: Just want to know the number of rainy days out of those three. Goal: Experiment: Observing the number of rainy days of the next three days on the Palouse. Set of possible outcomes (sample space), S: {0, 1, 2, 3}

14. Sample Space and Events Examples: Forecasting the weather the next three days on the Palouse: Some possible events (Possible subsets of S): • No rainy days will be observed = {0} Less than one rainy days will be observe = {<1} = {0} More than one rainy days will be observe = {>1} = {2, 3} • At least one rainy day will be observed = {>0} = {1, 2, 3}

15. Sample Space and Events Examples: Studying the chance of observing the faces of two dice when rolled: Observe faces of two dice Goal: Experiment: Rolling two dice Set of possible outcomes (sample space), S:

16. Sample Space and Events Examples: Studying the chance of observing the faces of two dice when rolled: 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)} • Die 2 will have face with number 3: {(1,3), (2,3), (3,3), (4,3), (5,3), (6,3)} • The simple event: Die 1 and Die 2 will have faces with numbers 3 and 4 respectively: {(3,4)}

17. Sample Space and Events Examples: Studying the chance of observing the sum of faces of two dice when rolled: Observe sum of faces of two dice Goal: Experiment: Rolling two dice Set of possible outcomes (sample space), S: {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}

18. Sample Space and Events Examples: Studying the chance of observing a smoker and observing that that smoker has lung cancer: Study association between smoking habits and lung cancer Goal: Experiment: Observe individuals and determine whether they smoke and have lung cancer or not Set of possible outcomes (sample space), S: {NN, SN, NC, SC}

19. Sample Space and Events Examples: Studying the chance of observing a smoker and then observing that that smoker has lung cancer: Some possible events (Possible subsets of S): • An individual is a smoker: {SN, SC} • An individual does not have cancer: {NN, SN} • An individual is a non-smoker and has cancer: {NC} All of the above examples had a finite sample spaces; i.e. we could count the possible outcomes.

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

21. Sample Space and Events Examples: Flipping a coin until the first head shows up: Some possible events (Possible subsets of S): • 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}

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

23. Sample Space and Events Examples: Flipping a coin until the first head shows up: 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} The above two examples had a countably infinite sample spaces; i.e. we could match the possible outcomes to the integer line.

24. Sample Space and Events Def. 1.2.2: A discrete sample space: is a sample space that is either finite or countably infinite

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

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