Chapter 6 Probability the study of randomness Probability- describes the pattern of chance outcomes Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run Random- if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Probability- is the proportion of times the outcome would occur in a very long series of repetitions. (long-term relative frequency) Independent trials- must not influence other trials 6.2 Probability models Probability models Sample Space S the set of all possible outcomes Event- is any outcome or set of outcomes (event is a subset of sample space)
Probability model mathematical description of a random phenomenon consisting of two parts: a sample space and a way of assigning probabilities to the events. Rolling 2 dice (sum) (Probability) (sum) (probability) 2 1/36 8 5/36 3 2/36 9 4/36 4 3/36 10 3/36 5 4/36 11 2/36 6 5/36 12 1/36 7 6/36 Probability of rolling a five (see notation below) P(5) (options 1-4; 4-1; 2-3; 3-2) P(5) = 4/36 or 1/9
Make a tree diagram Flip a coin and roll a die Coin Die Final Coin Die Final Outcome Outcome Outcome outcome outcome outcome 1 H1 1 T1 2 H2 2 T2 H 3 H3 T 3 T3 4 H4 4 T4 5 H5 5 T5 6 H6 6 T6 Multiplication principle One task a ways Second task b ways Then both a and b a b ways coin die (2) (6) 12 possible outcomes
Flip 4 coins 1 st flip 2 nd flip 3 rd flip 4 th flip (2) (2) (2) (2) 16 outcomes If we want to count only # of heads Outcomes are (0,1,2,3,4) heads, all heads) (no heads, one heads, two heads, three With replacement- when you draw, you put it back. Without replacement when you draw you do not put it back. Homework Read 330 340 do problems 11 15, 17, 18
Tuesday Probability rules 1) Any probability is a number between 0 and 1 2) All possible outcomes together must have probability 1 3) The probability that an event does not occur is 1 minus the probability that the event does occur. 4) If two events have not outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. In math terms 1.) A is an event 0 P(a) 1 2.) P(s) = 1 3.) Complement probability the event does not occur P(A c ) = 1 P(A) 4.) Disjoint (nothing in common) P(A or B) = P(A) + P(B) { A U B } A union B This means it is in A or B Empty Set Ø If disjoint or mutually exclusive then A B = Ø
Venn Diagram A B This is a Venn Diagram of a mutually exclusive or disjoint set. A U A c = S A A c = Ø Benford s Law Page 345 Homework Read pages 340 350 Do problems 19-23, 26
Wednesday Independence and multiplication Multiplication rule for independent events A and B are independent P(A and B) = P(A) P(B) Ex 6.14 page 353 Ex 6.15 Page 354 Homework read pages 354 355 do problems 27-29, 31
Thursday 6.3 General Probability rules Addition rule for disjoint events A, B, C have nothing in common P( one or more of A,B,C) = P(A) + P(B) + P(C) Addition rule for unions of two events P(A or B) = P(A) + P(B) P(A and B) Union means all areas P (A and B) = Ø A A & B B P(A) P(B) Ex 6.17 page 362 Deb.7 Matt.5 Together.3 Draw Venn diagram
Additional problem P(A) =.24 P(B) =.31 P(A and B) =.09 Draw picture Homework Read 364 365 Problems 46 53 Monday Conditional Probability P(A B) is conditional probability Read ex 6.18 page 366 Gives probability of one event under the condition we know the other event This symbol means given the information that
Do ex 6.19 pages 366 367 P(A and B) = P(A) P(B A) P(B A) = Ex 6.20 page 368 Homework read pages 369 371 do problems 54 61 Extended multiplication rules Intersection where all events occur (draw Venn diagram with three circles) P(A and B and C) = P(A)P(B A) P (C (A and B)) Tree diagram 5% go on to play at college level 1.7 % enter major league professional sports 40% have career more than 3 years
Professional sports.017 College.983 No professional sports.05 Male High School Athletes.95.0001 Professional sports Not College.9999 No professional sports P(Playing professional sports) (.05)(.017) + (.95)(.0001).00085 +.000095 =.000945 This means 9 out of every 10,000 high school athletes play professionally Bayes Rule If A and B are any events whose probability in not 0 or 1 P(A B) = A occurring given the information that B occurred
Two events A and B that both have positive probability are independent if P(A and B) = P(A)P(B) Diagram page 376 Homework do problems 62-64