1. AP Statistics
Notes Chapter 14 and 15

2. Notes Chapter 14
Probability is the branch of mathematics that describes the pattern of chance outcomes.
Random is a description of a kind of order that emerges only in the long run. Individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions.
The probability of any outcome is the proportion of times the outcome would occur in a very long series of repetitions.

3. For any random phenomenon, each attempt, or trial, generates an outcome.
The sample space, S, of a random phenomenon is the set of all possible outcomes for a variable. (Be sure you know what variable you are representing.)
Sampling can be done with or without replacement. Determining which to use is generally done from the context of the problem.
An event is an outcome or a set of outcomes of a random phenomenon, a subset of the sample space.

4. Disjoint events, also called mutually exclusive, have no outcomes in common. They can not occur at the same time.
Events are independent (informally) when the outcome of one event does not influence the outcome of any other event.
Be certain not to confuse disjoint with independent. Disjoint events cannot occur together, therefore they cannot be independent.

5. The Law of Large Numbers says that the long-run relative frequency of repeated independent events settles down to the true probability as the number of trials increases.
A probability distribution for a certain variable is made up of the sample space, S, and the individual probabilities of each event in the sample space.
The following three rules apply to probability distributions. S represents the sample space and x any event in the sample space.

6. Rule 1: The probability of any event is a number between 0 and 1
Rule 1: The probability of any event is a number between 0 and  P(x)  1.
Rule 2: The sum of all possible outcomes must
equal P(S) = 1.
Rule 3: The probability that an event does not
occur is 1 minus the probability that the event does occur.
This is called the probability of the complement, Ac.
P(Ac) = 1 – P(A)

7. An urn has 10 marbles in it. 4 are red, 3 are blue, 2 are green and 1 is purple.

8. Chapter 15
The union (U) of any collection of events is the event that at least one of the collection occurs. Union of two events is when either one or the other, or both events occur.
The intersection (∩) of any collect of events is when all the events occur. Intersection of two events is when both event A and event B occur.

9. Venn Diagrams:

10. Conditional Probability: P(B|A)
This is read as “the probability of B given A. It is the conditional probability that event B occurs given that event A occurs.

11. There are two rules of probability that are given on the AP Statistics formula sheet. The first is:
General Addition Rule for Union of Two Events
For any two events A and B,
P(A U B) = P(A) + P(B) – P(A ∩ B)
The second stems from the general multiplication rule which states:

12. General Multiplication Rule
P(A ∩ B) = P(B)P(A|B)
Based on the last formula we can define a rule for conditional probability:
OR….
When P(A) > 0,
P(A|B) = P(A ∩ B)
P(B)
This is the second formula for probability on the AP Stat formula chart.

13. Independent Events – formally
Two events A and B that both have positive probability are independent if
P(B|A) = P(B)
In other words, the probability of event B is not affected by the event A. This fits with the informal definition of independence but now gives us a way to show two events are independent mathematically. This formula is not on the AP Stat formula chart.

14. Making Tree Diagrams… Making Tables….

15. Example1: page 341 #24
The American Red Cross says that about 45% of the U.S. Population has Type O blood, 40% Type A, 11% Type B, and rest Type AB.
A) Someone volunteers to give blood. What is the probability that this donor
1. has Type AB blood?
2. has Type A or Type B?
3. is not Type O?

16. Example 1 continued
The American Red Cross says that about 45% of the U.S. Population has Type O blood, 40% Type A, 11% Type B, and rest Type AB.
B) Among four potential donors, what is the probability that
1. All are Type O?
2. no one is Type AB?
3. they are not all Type A?
4. at least one person is Type B?

17. Example 2: Birthday Gifts
In a book titles, Birthday Gifts, Brier and Rogers found that 71% of married men will get flowers or buy jewelry for their wives on their birthdays. 58% of married men will buy jewelry and 39% will buy flowers for their wives. What is the probability that a married man will buy flowers and jewelry for his wife’s birthday?

18. Example 3: Oh, Say can you see?
In the book Chances: Risk and Odds in Everyday Life, James Burke says that 56% of the general population wears eyeglasses, while only 3.6% wears contacts. He also says that of those who wear glasses, 55.4% are women. Of those who wear contacts, 36.9% are men. Assume that no one wears both glasses and contacts. For the next person you encounter at random, what is the probability that this person is:
A) not wearing contacts
B) a woman wearing glasses
C) a man wearing contacts

19. Example 4: Brown Hair/ Brown Eyes
In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. A person is selected at random from the town.
1. If he or she has brown hair, what is the probability that he or she also has brown eyes?
2. If he or she has brown eyes, what is the probability that he of she does not have brown hair?
3. What is the probability that he or she has neither brown hair or brown eyes?

20. Example 5: What does your GPA say?
This example does not fit on one slide, so we will have to use the document camera!!!