Probability: Understanding the likelihood of something happening.

Transcription

1 TED.MonaChalabi.adStatistics Probability: nderstanding the likelihood of something happening. Properties of probabilities They are numbers between 0 and 1 0 means the event under consideration is impossible 1 means the event under consideration is certain Sometimes expressed as a percent or as a fraction Not the same as odds Very commonly used in real world applications: 1/29 Intro 1

2 22: #Opener, 2,4 (Vocabulary, experimental probability) 1. nderstand and find experimental probability as a ratio of favorable to total outcomes. Experimental probability: The relative frequency of something occurring. (elative as compared to something else occurring.) Find the experimental probability of someone in the room having a birthday in February. Calculate irth Month Count Probability Jan Feb Mar pr May Jun Jul ug Sep Oct Nov Dec Total Probability often comes down to counting numbers of favorable outcomes and number of total outcomes. It's important to know what is what. 22: #Opener, 2,4 (Vocabulary, experimental probability) 1/29 22 Experimental Probability 2

3 22: #Opener, 2,4 (Vocabulary, experimental probability) Questions? 22: #1 3 (Sample space, Tree diagram) 22C.1: #3,5,6,7 (Theoretical probability, complementary events) 22C.2: #1 3 (rid diagrams) 1. Find and illustrate sample spaces using grid and tree diagrams Sample Space: The set of all possible outcomes of an experiment. Often denoted as. (as in niverse) What is the sample space for our experiment? epresentations of Sample Space: Lists Jan Feb Mar pr May Jun Jul ug Sep Oct Nov Dec irth Month Jan Feb Mar pr May Jun Jul ug Sep Oct Nov Tables Probability Tree Diagrams Jan Feb Mar pr May Jun Jul ug Sep Oct Nov Dec Dec rid Diagrams First Person (Experiments with 2 trials) Jan Feb Mar pr May Jun Jul ug Sep Oct Nov Dec J F M M J J S O N D Second Person Venn Diagrams (More on this later) 1/30 22 Sample Space 3

4 1. nderstand and calculate theoretical probability using symmetry and ratios. In some situations symmetry will define an expected probability. Consider: fair die fair coin spinner with equal (unequal?) areas fair die with sides labelled 1,1,1,2,3,4 The sum of the values when two dice are rolled (Craps!) The result of a oulette spin mong equally likely events, the mathematical or theoretical probability of an event is given by: Two events are complementary events if exactly one of them must occur. Examples: Heads & Tails oll a 6 and oll something other than a 6 (Not 6) The complement of is written as '. Some properties of theoretical probability: 0 P() 1 y definition P() + P(') = 1 Something must occur! 0 P 0.9e iπ To find theoretical probabilities, use areas, grids, and other ways of representing possible outcomes. rids and even cubes can be used to represent multiple events: Playing Yahtzee, after two rolls, you have 2,3,4,6,6. You need a large strait. Should you roll one of the 6's and hope for a 5 or should you roll both the sixes and hope for a 1 & 5 or a 5 & 6. P(rolling a 5 with one die) = 1/6 P(rolling a 1&5 or a 5&6 with two dice) = 4/36 = 1/9 etter to roll one die. 22: #1 3 (Sample space, Tree diagram) 22C.1: #3,5,6,7 (Theoretical probability, complementary events) 22C.2: #1 3 (rid diagrams) 1/30 22C Theoretical Probability 4

5 22: #1 3 (Sample space, Tree diagram) Present #3 22C.1: #3,5,6,7 (Theoretical probability, complementary events) Present #3,5 22C.2: #1 3 (rid diagrams) Present #3 1. nderstand and use tables of outcomes to calculate probabilities. 22D: #1 3 (Tables of outcomes) 22E.1: #1,3,5,6 (Compound events) 22E.2: #2 5 (Dependent events) s with statistics, variables involved in probability anaylsis are sometimes numeric (continuous or discrete) and sometimes categorical. ecall those definitions: Categorical Data Information that is described by different categories. Numerical or Quantitative Data Information that can be described by numbers. Numerical data can be broken into two subcategories Discrete Quantitative Data Information that takes on integer values. enerally, things that are counted. Continuous Quantitative Data Information that takes on real values. enerally, things that are measured. table can be helpful when there are two categorical variables to represent: Hair Color rown lack londe ed Men Women Interpreting these types of tables is not difficult, but be careful with the details. e sure you are calculating the correct numerator and denominator. Double check arithmetic by calculating totals in both directions. rown lack londe ed Total Men Women Total P(M) =? P(lack) =? P(W lack) =? P(M lack') =? Let's look at one more: 2/1 22D Tables of Outcomes 5

6 22C.1: #1,3,5,6,7 (Theoretical probability) 22C.2: #1 3 (sing grid diagrams) 22D: #1 3 (Tables of outcomes) Present questions 1. nderstand and interpret compound events Now for the fun stuff! Exploring the probability of two or more events occurring. Consider flipping a coin and rolling a die. What is the probability of getting a heads and a 5? heads or a 5? Independent Compound Events If the occurrence of event does not affect the occurrence of event (and vice versa) then and are called independent events. For independent events P( and ) = P() P() Similarly for more than two independent events: P( and and C) = P() P() P(C) Example: What is the probability of getting 3 bullseye's in a row on the following dart board (assume a blindfolded player with equal probability of landing anywhere in the large circle). 1 4 P(red) = rea(red)/rea(large) = π(1 2 )/π(5 2 ) = 1/25 = 0.04 P(3 in row) = (0.04)(0.04)(0.04) = Dependent Compound Events If the occurrence of event does affect the occurrence of event (or vice versa) then and are called dependent events. Consider a bowl with 5 red marbles and 5 green marbles. What is the probability of selecting a red then a green marble? P(ed) = 0.5 but then P(reen) = For dependent events P( then ) = P() P( given that occurred) Note that simultaneously drawing multiple objects from a hat creates dependent events. (Imagine reaching in, selecting on object, then the next, then the next etc. before withdrawing your hand.) When doing your HW, use proper probability notation to express your answers. For example: P()=0.3, P(S') = 0.4 or P( )=0.6 22D: #1 3 (Tables of outcomes) 22E.1: #1,3,5,6 (Compound events) 22E.2: #2 5 (Dependent events) 2/1 22E Compound Events 6

7 22E.1: #1 6 (Compound events) Present #1,3,4,5,6 22E.2: #1 4 (Dependent events) Present #1 4 When doing your HW, use proper probability notation to express your answers. For example: P()=0.3, P(S') = 0.4 or P( )= nderstand and interpret tree diagrams 22F: #1 8 (Tree diagrams) Q: #1,2,3,8,16 (I Practice) Very powerful and will be even more useful as we proceed! Consider rolling a die, then flipping a coin (independent events) H P(1 then H) = T P(1 then T) = H P(2 then H) = T P(2 then T) = H P(3 then H) = T P(3 then T) = H P(4 then H) = T P(4 then T) = H P(5 then H) = T P(5 then T) = H P(6 then H) = T P(6 then T) = The probability of a particular sequence is the product of the probabilities along that path. P(M C) P(C M') Tree diagrams are exceptionally useful! When confused, try drawing a tree diagram. 22F: #1 8 (Tree diagrams) Q: #1,2,3,8,16 (I Practice) Note: e thorough with the HW. These problems begin the transition to more sophisticated ideas. nderstand the problems. 2/5 22F Tree Diagrams (Independent) 7

8 22F: #1 8 (Tree diagrams) Present #3,4,5,7 Q: #1,2,3,8,16 (I Practice) Present #3,8,16. Try #9! In the problem below, please leave all answers as unreduced fractions. In the Powerball lottery five numbers between 1 and 69 are selected along with one "Powerball" number between 1 and 26. The initial five numbers can be in any order and are not repeated. The Powerball number is chosen separately and can be one of the initial What is the probability that the first number on your ticket matches the first number drawn? [2] 2. What is the probability that the second number on your ticket matches the first number drawn? [2] 3. What is the probability that one of the five numbers on your ticket matches the first number drawn? [2] 4. Suppose one of your numbers matches the first number drawn. What is the likelihood that one of your remaining numbers matches the second number drawn? [2] 5. What is the probability that the five numbers on your ticket match the five initial numbers drawn? [2] 6. What is the probability of winning the jackpot matching all five numbers and the Powerball number?[2] 1) 1/69; 2) 1/69; 3) 5/69; 4) 4/68; 5) 120/1,348,621,560; 6) 1/292,201,338 Imagine lining up baseballs ( standard baseball is about 2.9 inches in diameter.) in a row for the 2, highway miles from oston to Los ngeles. It would take about 65,515,988 baseballs. Then randomly designate one of these baseballs as a lucky winner baseball. Imagine driving for days past this row of millions and millions of baseballs. Then stop and pick up a random baseball. The chance of a random ticket winning the new Powerball is less than one fourth the chance of picking the winning baseball. 22: #4 10 (Sampling with and without replacement) Q: #5,6,9,10 (Sampling w/wout replacement) 1. nderstand and solve problems involving probabilities of dependent events Tree Diagrams are very useful for dependent events. Consider a box with 5 green, 5 red, and 5 blue balls. You select three balls without replacement. Sampling without replacement creates dependent events. Probability of a particular sequence is the product along the paths 22: #4 10 (Sampling with and without replacement) Q: #5,6,9,10 (Sampling w/wout replacement) 2/6 22 Sampling without replacement 8

9 22: #4 10 (Sampling with and without replacement) Present 8,10 Q: #5,6,9,10 (Sampling w/wout replacement) Present 6,9,10 Quick Quiz (no calc) (a.0) Draw a tree diagram illustrating the situation. (4) (a.0) 1 for correct tree 111 for probabilities.1 1. nderstand basic ideas used in Venn Diagrams Venn Diagrams 22H.1: #3 8,10,11 (Sets & Venn diagrams) 22H.2: #1 3 (Set identities & notation) Q: #11,12,14 (Q practice) way to represent the relationship between events. Sample space is drawn as a rectangle (or the outermost oval) Events of interest are drawn with circles and ovals containing the favorable outcomes of that event. Overlapping regions indicate that an outcome is part of multiple events. reas are not drawn proportionally to the number of outcomes inside ' 2,4,6 1,3, 5 etting an even roll of a die The notation n() represents the number of elements in set. So in this case n() = n(') = 3 Set Notation The following notation is taken from the HL I notation sheet = {x x or x or x } (This is an inclusive or) = {x x and x } = Disjoint sets (Mutually exclusive) Venn diagrams can help you understand certain types of problems. 2/8 22H.1 Sets & Venn Diagrams 9

10 .2 1. se set notation and algebra to work with Venn diagrams. Can we prove some set Identities? So notice that P(' ') = P(( )') = 1 P( ) So notice that P(' ') = P(( )') = 1 P( ) You try some with 3 sets: Kind of like distributing... Conjecture: Set operations are commutative but not associative. Can you prove this? e very thorough with the HW You are building foundations 22H.1: #3 8,10,11 (Sets & Venn diagrams) 22H.2: #1 3 (Set identities & notation) Q: #11,12,14 (Q practice) 2/8 22H.2 Proving Set Identities 10

11 22H.1: #3 8,10,11 (Sets & Venn diagrams) Present #7,10,11 22H.2: #1 3 (Set identities & notation) Present #3 Q: #11,12,14 (Q practice) Present all Consider: 2I: #1 3,6 14 even (Laws of probability, conditional probability) 1. nderstand and use the addition and conditional laws of probability Probability nit Exam Next Thu, 2/22 What is the probability that the student likes either pizza or coffee? Draw a Venn Diagram Notice that when we add n(p) + n(c) ( ) we are counting 6 students twice. To find our answer, we can subtract: n(p) + n(c) n(p C) In general, then the ddition Law of Probability states that:...or in words... Note that for disjoint events, P(both and ) = 0. So for this case Is the converse true? Can you prove it? Try one: In a class of 40 students, 34 like bananas, 22 like pineapples, and 2 dislike both. What is the probability that a student likes either bananas or pineapples? Conditional Probability means that occurs, given that has already occurred. lso written as given. What is the probability that a student who likes pizza also likes coffee? efer to the Venn Diagram If we know they liked pizza, then there are only 14 people in the sample space. Of those, 6 like coffee. So the answer is lgebraically: Probability of liking coffee given liking pizza Probability of liking both Sample space is reduced because we know they like pizza enerally the Conditional Law of Probability states that: Notice that the position in the fraction is very important! nderstand why! Sometimes it's more intuitive to use another form of this result: Notice that you can calculate the intersection from either conditional. Try one: This HW is very important. It collects all the ideas we've seen in a more abstract way. Do it thoroughly. 2I: #1 3,6 14 even (Laws of probability, conditional probability) Probability nit Exam Next Thu, 2/22 2/13 22I Laws & Conditional 11

12 22I: #1 3,6 14even (Laws of probability, conditional probability) Present 6 14 even 1. nderstand and use the addition and conditional laws of probability 22J: #1 8 (Independent events) Q: #13,15 (Independence) Probability nit Exam Next Thu, 2/22 Note: In the Q the symbol means '. Last of the content time to synthesize. nit Exam Thu 3/9 Our previous definition If the occurrence of event does not affect the occurrence of event (and vice versa) then and are called independent events. lgebraically this implies P( ) = P() (:"I won $1M dollars!" : "I don't care. I like vanilla ice cream.") P( ') = P() (:"I lost $1M dollars!" : "I'm sorry. I still Iike vanilla ice cream.") We can rearrange the conditional law of probability to reveal another interesting idea: ecall that P( ) = P( )/P() always. lternatively: P( ) = P( )P() always. If and are independent events P( ) = P() and: P( ) = P() P() for independent events Note that the converse is also true. This allows us to answer questions like: is getting heads on a coin, is getting less than 3 on a die. Show that and are independent. sing a grid or tree diagram, we can show that Therefore, and are independent. nother observation: efer back to the addition law of probability: P( ) = P() + P() P( ) = P() P() Û indep ut when and are independent, the last term is just P() P(). Thus =0 Û disjoint and are independent events Û P( ) = P() + P() P() P() Summary of Properties of Independent Events and are independent events means P( ) = P( ') = P() and are independent events Û P( ) = P() P() and are independent events Û P( ) = P() + P() P() P() Let's do some problems that these ideas illuminate: fter #13 & 15, you should have completed the entire set of Q problems Coming up: eview Tue 2/20 nit Exam on Thursday, 2/22. 22J: #1 8 (Independent events) Q: #13,15 (Independence) Probability nit Exam Next Thu, 2/22 2/15 22J Independent Events 12

13 22J: #1 8 (Independent events) Present all Q: #13,15 (Independence) Present all Exploration: Detailed intro and handouts Main Probability Laws ddition Law of Probability Complements and Set Identities P(')= 1 P() P( ) = P() + P() P( ) P( C) = P() + P() + P(C) P( ) P( C) P( C) + P( C) = (' ')' = (' ')' ( ) ( ') = ' ' = ( )' ' ' = ( )' Conditional Law of Probability ayes' Theorem Special Cases Independent Events P( ) = P()P() P( ) = P( ') = P()...so P( ) = P() + P() P()P() Disjoint Events (Mutually exclusive) P( ) = 0 P( ) = 0...so P( ) = P() + P() eview assignments: ll questionbank problems should be completed. Tue 2/20 Q: eview the ones you've done. eview: #Set 22,, & C (Do a selection 3 from each section. Choose the ones that challenge you.) Thu 2/22: Probability nit Exam 2/20 eview & Summary of Formulae 13