Sample Space and Events

1. Sample Space and Events Section 2.1 An experiment: is any action, process or phenomenon whose outcome is subject to uncertainty. An outcome: is a result of an experiment. Each run of an experiment results in only one outcome! A sample space: is the set of all possible outcomes, S, of an experiment. An event: is a subset of the sample space. An event occurs when one of the outcomes that belong to it occurs. A simple (or elementary) event: is a subset of the sample space that has only one outcome.

2. Still in Ch2: 2.2 Axioms, interpretations and properties of probability

3. Axioms, interpretation and properties Section 2.2 Probability: is a measure of the chance that an event might occur (before it really occurs). Philosophically: Probability (frequentist): the limiting relative frequency of the occurrence of an event when running the associated experiment over and over again for a very long time, which gives us an indication about the chance of observing that event again when running the experiment one more time (followed in this book) Probability (Bayesian): A measure of belief (objective or subjective) of the chance that an event might occur.

4. Axioms, interpretation and properties Section 2.2 Example: Flipping a fair coin. Frequentist: If we flip a physically balanced (fair) coin over and over again for a long time then the proportion of times that one will observe a head will be ½ of the time. This assumes that air friction and other factors are either controlled or negligible. Bayesian: If a coin is physically balanced (fair) and if the effect of air friction and other factors that might affect its orientation when landing are all either negligible, or controlled for, then the chance that a head is observed when that coin is flipped is ½.

5. Axioms, interpretation and properties Section 2.2 Mathematically Probability: is a function that uses events as an input and results in a real number between 0 and 1 as an output that describes the chance that an event might occur. Because it uses events as input and because events are sets (they are subsets of the sample space) a probability function is said to be a set function.

6. Axioms, interpretation and properties Section 2.2 A probability function must obey some rules for it to make sense mathematically and logically. These rules are called axioms

7. Axioms, interpretation and properties Section 2.2 Axioms (rules) of probability: For any event A, Probability of observing the sure event, S, if you run the experiment is: If A1, A2, A3, …is an infinite collection of disjoint events (i.e. for any i and j where ), then

8. Axioms, interpretation and properties Section 2.2 Axioms (rules) of probability: We can also show that if A1, A2, A3, … An is a finite collection of disjoint events (i.e. for any i and j where ), then

9. Axioms, interpretation and properties Section 2.2 Examples: Studying the blood types of a randomly sampled individual: Observe an individual’s blood type Goal: Experiment: Choose an individual at random and observe his/her blood type. Set of possible outcomes (sample space), S: {O, A, B, AB} Events (subsets of S): (16 of them)

10. Axioms, interpretation and properties Section 2.2 Examples: Studying the blood types of a randomly sampled individual: Let E be the simple event {O}, is the following correct? P(E) = -0.5

11. Axioms, interpretation and properties Section 2.2 Examples: Studying the blood types of a randomly sampled individual: Let E be the simple event {O}, C be the event {A, AB} and D the event {O, A, AB}, is the following correct? P(E) = 0.2 P(C) = 0.3 P(D) = 0.4

12. Axioms, interpretation and properties Section 2.2 Examples: Studying the blood types of a randomly sampled individual: Let E1 be the simple event {O}, E2be {A}, E3 be {B} and E4 be {AB}, is the following correct? P(E1) = 0.2 P(E2) = 0.3 P(E3) = 0.4 P(E4) = 0.4

13. Axioms, interpretation and properties Section 2.2 Examples: Studying the blood types of a randomly sampled individual: We can construct a base probability distribution (the probability distribution) associated with this example using the simple events as follows: P(E1) = 0.2 Or P(E2) = 0.1 P(E3) = 0.4 Or P(E4) = 0.3

14. Axioms, interpretation and properties Section 2.2 Examples: Studying the blood types of a randomly sampled individual: We can use this probability distribution to find the probability of any event:

15. Axioms, interpretation and properties Section 2.2 Examples: Studying the chance of observing the faces of one fair die when rolled: Observe faces of one die Goal: Experiment: Rolling a die Set of possible outcomes (sample space), S: {1, 2, 3, 4, 5, 6} Events (subsets of S): (64 of them)

16. Axioms, interpretation and properties Section 2.2 Examples: Studying the chance of observing the faces of one fair die when rolled: Probability distribution: This is a special case where each of the simple events are equally likely. For any event A, P(A) in this case is given by:

17. Axioms, interpretation and properties Section 2.2 Some other properties: For any event A: P(A)+P(A’) = P(S) = 1 => P(A) = 1 – P(A’) Note that and Now, Axioms 2 and 3 will help us show this rule. A’ S A

18. Axioms, interpretation and properties Section 2.2 Some other properties: The probability of the empty set is: Note that , and Now, property (1) will help us show this property.

19. Axioms, interpretation and properties Section 2.2 Some other properties: For any events A and B where : S B A

20. Axioms, interpretation and properties Section 2.2 Some other properties: For any event A: Note that and use property (3)

21. Axioms, interpretation and properties Section 2.2 Some other properties: For any events A and B: S A B

22. Axioms, interpretation and properties Section 2.2 Some other properties: Example: 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 if weather is rainy (R) or not (N) in each of those days. Set of possible outcomes (sample space), S: {NNN, RNN, NRN, NNR, RRN, RNR, NRR, RRR}

23. Axioms, interpretation and properties Section 2.2 Some other properties: Example: Forecasting the weather for each of the next three days on the Palouse: Let A be the event that day 3 is rainy, P(A) = 0.4 Let B be the event that day two is rainy, P(B) = 0.7 Let C be the event days 2 and 3 are rainy, P(C) = 0.3 Find probability that either day 2 or 3 will be rainy.

24. S A NNR, RNR B NRN, RRN C NRR, RRR

25. Axioms, interpretation and properties Section 2.2 Some other properties: Example: Forecasting the weather for each of the next three days on the Palouse: Let A be the event that day three is rainy and P(A) = 0.4 Let B be the event that day two is rainy and P(B) = 0.7 Let C be the only days 2 and 3 are rainy and P(C) = 0.3 Find probability that either day 2 or 3 will be rainy. = 0.4 + 0.7 – 0.3 = 0.8

26. Axioms, interpretation and properties Section 2.2 Some other properties: Example: Forecasting the weather for each of the next three days on the Palouse: Let A be the event that day 3 is rainy, P(A) = 0.4 Let B be the event that day two is rainy, P(B) = 0.7 Let C be the event days 2 and 3 are rainy, P(C) = 0.3 Find probability that day 3 and not day 2 is rainy.

27. Axioms, interpretation and properties Section 2.2 Some other properties: Example: Forecasting the weather for each of the next three days on the Palouse: Let A be the event that day 3 is rainy, P(A) = 0.4 Let B be the event that day two is rainy, P(B) = 0.7 Let C be the event days 2 and 3 are rainy, P(C) = 0.3 Find probability that day 3 is not rainy.