Friday Feb 23, is the deadline for registering and paying for AP Exams Friday Feb 23, is the deadline for registering and paying for AP Exams. If you have not taken care of this, you need to do it immediately. It is a red flag to college admissions offices if students register for Advanced Placement classes but choose to not take the national exam. Receiving college credit is the purpose for taking AP classes.

Advanced Placement Statistics Section 6.2: Probability Models EQ: How do we use probability models to determine the chance of a specific event occurring?

Get out your textbook. Terms to Know: Probability Model --- mathematical representation of a random phenomenon; defined by a sample space, events in the sample space, and the probabilities of these events occurring Sample Space --- collection of all possible outcomes Get out your textbook.

In Class p. 416 #29 a) S = {germinates, fails to grow} Survival time could be measured in days, weeks, months, years, etc. c) S = { A, B, C, D, F}

RECALL: Tree Diagram ---

Back to p. 416 #29 d) Let H = hit Let M = miss S = {HHHH,HHHM,HHMH,HHMM,….MMMM} There are 16 elements in this sample space. e) S = {0, 1, 2, 3, 4}

DAY 35 AGENDA: D16 --- 20 minutes

Event --- desired outcome Probability of an Event --- P(E) = # of outcomes in event # of outcomes in sample space = n(E) n(S)

Ex. Match the probabilities that follow with each statement about an event. 0.0 0.01 1.0 0.3 0.6 0.99 1.4 This event is unlikely but it will occur once in a while in a long sequence of trials. This event will occur more often than not. I don’t know what this number represents, but it isn’t probability. This event is certain. It will occur on every trial of the random phenomenon. This event is impossible. It will never occur. 0.01 0.6 1.4 1.0 0.0

Multiplication Principal --- Fundamental Counting Principle If there are m ways to make a first selection and n ways to make a second selection, there are m × n ways to make the two selections. Ex. How many outcomes can occur if you flip a coin and toss a die? State the sample space. 2 x 6 = 12 S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} ***Recall p. 416 #29

Venn Diagram --- shows all possible logical relations between a finite collection of sets.

Union of Two Sets --- all elements in either A or B

Intersection of Two Sets --- elements found in both A and B

Complement of a Set --- all of the elements in the universe that are not in the set.

Emperical vs Theoretical based on the results of an experiment or simulation number of ways an event can occur based on the events in the sample space of what should happen, not what did happen

Example: Let U (the universal set) ={1, 2, 3, 4, 5, 6, 7, 8, 9,10} (a subset of the positive integers) Given: A = {2, 3, 5, 7, 8} B = {1, 2, 4, 6, 7} Find: A U B =   A ∩ B = (A U B)c =         Ac ∩ B c = {1, 2, 3, 4, 5, 6, 7, 8} {2, 7} {9, 10} {9, 10}

Ex. Create a Venn Diagram for the following   Thirty males at OCHS were asked whether they played football, basketball, and/or baseball. Four responded that they played all three sports, 19 said they played football, 3 said they played football and basketball, 13 said they played baseball of which 4 said they played only baseball, no one responded that they played basketball and baseball, and 5 males said they do not play sports.

Ex. Create a Venn Diagram for the following. Twenty-four dogs are in a kennel.  Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair.  There is only one dog that is black with a short tail and long hair.  Two of the dogs are black with short tails and do not have long hair.  Two of the dogs have short tails and long hair but are not black.  If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails?

Ex. Create a Venn Diagram for the following. Twenty-four dogs are in a kennel.  Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair.  There is only one dog that is black with a short tail and long hair.  Two of the dogs are black with short tails and do not have long hair.  Two of the dogs have short tails and long hair but are not black.  If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails? (15 – x – 3) + (12 – x – 3) + x + 6 = 24 12 – x + 9 – x + x + 6 = 24 -x = -3 x = 3 3 dogs are black with long hair and do not have short tails.

 MUTUALLY EXCLUSIVE INDEPENDENCE Disjoint Events (aka Mutually Exclusive Events) --- 2 events which cannot happen at the same time MUTUALLY EXCLUSIVE  INDEPENDENCE

Two Scenarios to Compare: Independence vs Mutually Exclusive Ex. 1 I agree to pay you $10 if you correctly guess the color of the card I have drawn from the deck. You do not hesitate, take a 50-50 chance and say red. The probability you win $10 is 50%.   Suppose before you guess, I take a peek at the card and give you a hint.

Scenario 1: I tell you the card is a spade Scenario 1: I tell you the card is a spade. Since there are no red spades, these events are mutually exclusive. My hint guarantees you $10. Scenario 2: I tell you the card is an ace. That’s no help, since there’s still a 50-50 chance its red (2 out of 4 suits). My hint gave you no useful information. Therefore these events are independent.

Events don’t have to be independent or mutually exclusive. Ex. Colored Blindness Colorblindness is a gender-linked trait, it’s predominantly (but not entirely) men who are colorblind. Consider the events “Female” and “Colorblind”.  Since some women are colorblind, these events are not mutually exclusive. But they are not independent either, as the rate of colorblindness among females is lower. (Gender does have some effect on colorblindness.)

Formulas to Know for Calculating Probability: Intersection of Two Independent Events P(A  B) = P(A)  P(B) Complement of an Event   P(Ac) P(A’) = 1 – P(A) Probability of Two Disjoint Events   P(A  B) = P(A) + P(B) Probability of Two Events that are NOT Disjoint   P(A  B) = P(A) + P(B) – P(A  B)

Therefore the events are NOT INDEPENDENT Ex. Use the following table to answer these questions. 0.40 0.60 0.42 0.58 1.00 Find the probability “read text” and “did not read novel”.   P(T  N ’) = 0.1 Are these events INDEPENDENT? RECALL Two Independent Events are Independent if P(T  N’) = P(T)  P(N’) Proof: DOES .1 = (.4)(.58) ? .1  .232 Therefore the events are NOT INDEPENDENT

Therefore the events are NOT INDEPENDENT Ex. Use the following table to answer these questions. b) Find the probability “read novel” and “did not read text”.  0.12 P(N  T ’) = Are these events INDEPENDENT? RECALL Two Independent Events are Independent if P(N T ’) = P(N)  P(T ’) PROOF: DOES .12 = (.42)(.6) ? .12  .252 Therefore the events are NOT INDEPENDENT

Ex. Use the following table to answer these questions. c) Make a Venn Diagram indicating the events A = “read text” or B = “read novel”.   0.10 0.30 0.12 0.48

Ex. The following table represents the proportion of women aged 25 to 29 who have that marital status. Show two ways to find P(not married).   Find P(never married or divorced). P(NM U W U D) = P(NM) + P(W) + P(D) = .353 + .002 + .071 = .426 or 1 - P(M) = 1 - .574 = .426 P(NM U D) = P(NM) + P(D) = .353 + .071 = .424 NOTE: These events are mutually exclusive (disjoint). There is NO INTERSECTION.

Assignment p. 417 #33, 35, 36 p. 423 – 425 #37 – 41, 44 p. 430 – 431 #45 - 52