Section Probability Models AP Statistics December 2, 2010

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

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

4 Sample Space: Rolling 2 Dice 1,21,31,41,51,6 2,22,32,42,52,6 3,23,33,43,53,6 4,24,34,44,54,6 5,25,35,45,55,6 6,26,36,46,56,6 1,1 2,1 3,1 4,1 5,1 6,

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

6 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.

7 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

8 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

9 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? *** = 5040 ways

10 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

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

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

13 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…? 14

15 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.

16 Probability Rules: Rule 3 (Different Notation) If (A B) = Ø, P (A B) = P (A) + P (B) = AND = 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 P(club or diamond) = 26/52 = 1/2 17

18 Probability Rules: Rule 4 The complement of any event A is the event that A does not occur, written as A C. The complement rule states that P (A C ) = 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 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. Age Group (yr)18 to 2324 to 2930 to 3940 or over Probability P(18 to 23) =.57 1 – P(18 to 23) =1 –.57 =.43 P c (18 to 23) = P(at least 24) =P(24 to 29) +P(30 to 39) +P(40 or over) = = =.43

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

AP Statistics, Section 6.2, Part 3 21 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 22 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 23 Example of Not Independent Events Choosing two cards without replacement Scoring above 600 on verbal SAT, scoring 600 on math SAT

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 24

AP Statistics, Section 6.2, Part 3 25 Independent and complements If A and B are independent, then so are…  A c and B c  A and B c  A c and B

AP Statistics, Section 6.2, Part 3 26 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 27 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 28 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) .6 6 = P(getting through all 6 lights) .4 6 =