Section 6.2.1 Probability Models

Section 6.2.1Probability Models AP Statistics December 2, 2010

Sample Space • The sample space S of random phenomenon is the set of all possible outcomes.

Sample Space {girl, boy} • For a table of random digits it is S = _______________. {0,1,2,3,4,5,6,7,8,9} For a flipped coin, the sample space is S = {H, T}. For a child's sex it is S =_________.

Sample Space: Rolling 2 Dice

Sample Space: Flipping a Coin and Rolling a Die **Tree Diagram!**

Probability Model • A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events.

Multiplication Principle • If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a·b number of ways. • Example: You flip a coin and then roll a die. How many possible outcomes are there in the sample space? • (# possible outcome coin flip) * (# possible outcomes die roll) = total possible outcomes • 2 * 6 = 12 possible outcomes in S

Multiplication Principle • Sampling WITH replacement: • If you draw a slip of paper from a hat with 10 slips. You replace the paper each time. How many possible outcomes are in your sample space if you draw a slip 4 times? • 10*10*10*10=10,000 possible outcomes in the sample space

Multiplication Principle • Sampling WITHOUT replacement: • If you draw a slip of paper from a hat with 10 slips. You DO NOT replace the paper each time. How many possible outcomes are in your sample space if you draw a slip 4 times? 10 9 8 7 = 5040 ways * * *

Event • An event is any outcome or a set of outcomes of a random phenomenon. • That is, an event is a subset of the sample space

Notation Read P(A) as “the probability of event A”

Probability Rules: Rule 1 The probability P(A) of any event A satisfies 0 ≤ P (A) ≤ 1

Probability Rules: Rule 2 If S is the sample space in a probability model, then P(S) = 1 In other words, the sum of probabilities of all possible outcomes must equal 1.

Apply Probability Rule #2 Choose a STATS AP student at random. P(student has blonde hair) = .15, P(student has brown hair) = .6, P(student has black hair) = .2, P(student has red hair) = .1 What’s wrong…?

Probability Rules: Rule 3 • Two events A and B are disjoint (also called “Mutually Exclusive” if they have no outcomes in common and so can never occur simultaneously. • EX: drawing a club or drawing a diamond • If A and B are disjoint, P (A or B) = P (A) + P (B). This is the addition rule.

Probability Rules: Rule 3 (Different Notation) = AND • If (A B) = Ø, • P (A B) = P (A) + P (B) = OR This is the addition rule for disjoint events

Apply Probability Rule #3 What is the probability of drawing a club or drawing a diamond? P(club or diamond) = P(club) + P(diamond) P(club or diamond) = 13/52 + 13/52 P(club or diamond) = 26/52 = 1/2

Probability Rules: Rule 4 The complement of any event A is the event that A does not occur, written as AC. The complement rule states that P (AC) = 1 - P (A). “The probability that an event does not occur is 1 minus the probability that the event does occur.”

Apply Probability Rule #4 • P(18 to 23) = .57 • Pc(18 to 23) = 1 – P(18 to 23) = 1 – .57 = .43 • P(at least 24) = P(24 to 29) +P(30 to 39) +P(40 or over) = = .17 + .14 + .12 = .43 Distance learning courses are rapidly gaining popularity among college students. Below is a probability model showing the proportion of all distance learners in each age group.

Exercises: 6.29, 6.32, 6.33, 6.36, 6.38, 6.41, 6.44

Definition of Independence Two events A and B are independent if knowing that one occurs does not change the probability of that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) This is the multiplication rule for independent events AP Statistics, Section 6.2, Part 3

Example of Independent Events • First coin flip, second coin flip • Rolling of two dice • Choosing two cards with replacement AP Statistics, Section 6.2, Part 3

Example of Not Independent Events • Choosing two cards without replacement • Scoring above 600 on verbal SAT, scoring 600 on math SAT AP Statistics, Section 6.2, Part 3

Probability Rule #5: Multiplication Rule for Independent Events If two events A and B are independent, then P(A and B) = P(A)P(B) EX: What is the probability of rolling a die and getting an odd, then a three? P(odd and 3) AP Statistics, Section 6.2, Part 1

Independent and complements • If A and B are independent, then so are… • Ac and Bc • A and Bc • Ac and B AP Statistics, Section 6.2, Part 3

Are these events independent? • A={person is left-handed} • B={person is an only child} • C={person is blue eyed} AP Statistics, Section 6.2, Part 3

Are these events independent? • A={person is college graduate} • B={person is older than 25} • C={person is a bank president} AP Statistics, Section 6.2, Part 3

Traffic light example • Suppose the timing of the lights on morning commute are independent. • The probability of being stopped at any light is .6. • P(getting stopped at all the lights) • .66=.046656 • P(getting through all 6 lights) • .46=.004096 AP Statistics, Section 6.2, Part 3

Section 6.2.1 Probability Models