1. 1 Probability: Liklihood of occurrence; we know the population, and we predict the outcome or the sample. Statistics: We observe the sample and use the statistics to describe the unknown population. When we make an inference about the population, it is desirable that we give a measure of ‘confidence’ in our being correct (or incorrect). This is done giving a statement of the ‘probability’ of being correct. Hence, we need to discuss probability. Probability 2. More terms:

2. 2 So, Probability: more terms: event: is an experiment y: the random variable, is the outcome from one event sample space: is the list of all possible outcomes probability, often written P(Y=y), is the chance that Y will be a certain value ‘ y ’ and can be computed: number of ways to succeed p = --------------------------------. Sample Space

3. 3 8 things to say about probability: 1. 0  p  1 ; probabilities are between 0 and 1 inclusive. 2.  p i = 1 ; the sum of all the probabilities of all the possible outcomes is 1. Event relations 3. Complement: If P(A=a) = p then A complement is is P( A not a) = 1-p (= q sometimes). Note: p + (1-p) = 1 (p+q=1). A complement is also called (not the mean).

4. 4 8 things continued 4. Mutually exclusive: two events that cannot happen together. If P(AB)=0, then A and B are M.E. 5. Conditional: Probability of A given that B has already happened. P(A|B) 6. Independent: Event A has no influence on the outcome of event B. If P(A|B) = P(A) or P(B|A) = P(B) then A and B are independent.

5. 5 8 things continued again two laws about events: 7. Multiplicative Law: (Intersection ; AND) P(AB) = P(A  B) = P(A and B) = P(A) * P(B|A) = P(B) * P(A|B) if A and B are independent then P(AB) = P(A) * P(B). 8. Additive Law: (Union; OR) P(A  B) = P(A or B) = P(A) + P(B) - P(AB).

6. 6 A B AB The Venn diagram below can be used to explain the 8 ‘things’.

7. 7 An example: Consider the deck of 52 playing cards: sample space: (A, 2, 3, …, J, Q, K) spades (A, 2, 3, …, J, Q, K) diamonds (A, 2, 3, …, J, Q, K) hearts (A, 2, 3, …, J, Q, K) clubs Now, consider the following events: J= draw a J: P(J)= 4/52=1/13 F = draw a face card (J,Q,K): P(F)= 12/52=3/13 H = draw a heart: P(H)= 13/52

8. 8 An example.2: Compute the following: 1. P(F complement) = (52/52 - 12/52) = 40/52 2. Are J and F Mutually Exclusive ? No: P(JF) = 4/52 is not 0. 3. Are J and F complement M.E. ? Yes: P(J and ) = 0 4. Are J and H independent ? Yes: P(J) = 13/52 = 1/13 = P(J|H)

9. 9 An example.3: Compute the following: 5. Are J and F independent ? No: P(J) = 4/52 but P(J|F) = 4/12 6. P(J and H) = P(J) * P(H|J) = 4/52*1/4 = 1/52 7. P(J or H) = P(J) + P(H) - P(JH) = 4/52 + 13/52 - 1/52 = 16/52. End: Probability 2.