1. PROBABILITY THEORY RULES OF PROBABILITY - review Rule 1: 0  P(A)  1 for any event A –Any probability is a number between 0 and 1 Rule 2: P(S) = 1 –All possible outcomes together must have a probability of 1 –Some outcome must occur on any trial Rule 3: For any event A, P(A does not occur) = 1 – P(A) Rule 4: Addition rule: If A and B are disjoint events, then P(A and B) = P(A) + P(B)

2. Example –All human blood can be typed as one of O, A, B, or AB, but the distribution of the types varies a bit with race. Here is the probability model for the blood type of a randomly chosen black American: Blood typeOABAB Probability 0.49 0.27 0.20 ? A: What is the sample space of blood type for black Americans? B: What is the probability of AB type? Why? C: What is the probability of type O or A? D: What is the probability that an individual is not type O? E: What is the probability that an individual has the substance A in his/her blood? F: Maria has type B blood. She can safely receive blood transfusions from people with blood types O and B. What is the probability that a randomly chosen black American can donate blood to Maria?

3. PROBABILITY THEORY When the outcome of one event does not affect or predetermine the outcome of another event, we say that the two events are INDEPENDENT. EXAMPLE 2: A standard 52-deck of cards contains 26 red, and 26 black cards. For the first card dealt from a shuffled deck, the probability of a red card is _______. Given that a red card is drawn and removed from the deck, what is the probability of drawing a red card on second trial? Are these events independent? A: draw a red card on first try B: draw a red card on second trial

4. PROBABILITY THEORY VENN DIAGRAM Diagram that displays a sample space and events within it Depicts relationships among events Fig 5.1 Fig 5.2

5. PROBABILITY THEORY MULTIPLICATION RULE FOR INDEPENDENT EVENTS Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. For two such events P(A and B) = P(A). P(B)

6. PROBABILITY THEORY EXAMPLE: Select a first-year college student at random and ask what his or her academic rank was in high school. Here are the probabilities based on proportions from a large sample survey of first-year students: A: Choose two first-year college students at random. Why is it reasonable to assume that their high school ranks are independent. B: What is the probability that both were in the top 20% of their high school classes? C: What is the probability that the first in the top 20% and the second was in the lowest 20%? RankTop 20% Second 20% Third 20% Fourth 20% Lowest 20% Probability0.410.230.290.060.01

7. Probability Intervals of outcomes –P(0.4 ≤ X ≤ 0.8) =_____________ –P(X ≤ 0.5) = _________________ –P(X > 0.8) = _________________ –P(X ≤ 0.5 or X > 0.8) =_________ A = 1 1 1 0

8. Probability Normal Probability Distributions –Any density curve can be used to assign probabilities –Normal distribution – probability model –Heights of all young women follow a normal distribution with μ of 65.5 inches and σ if 2.5 inches. –This is a distribution for a large set of data. –If you choose any woman C at random, and repeat the randomization many times, the distribution of values of C follow normal distribution.

9. Normal Probability Distributions Example: –What is the probability that a randomly chosen young woman had height between 68 and 70 inches? Distribution of Height =N=(64.5, 2.5) Remember – Z table finds probabilities of standardized data.

10. Probability Example –The random variable X has the standard normal N(0,1) distribution. Find each of the following probabilities: A: P(-1 ≤ X ≤ 1) B: P(1 ≤ X ≤ 2) C: P(0 ≤ X ≤ 2)