Introduction to Simulations and Experimental Probability Chapter 4.1 – An Introduction to Probability Learning goals: Describe a simulation for a given scenario Calculate experimental probability Read through Example 2 - solution 1, p. 207 Complete pp #1, 5, 8, 9-10, 12-13

Investigation – Experimental Probability Work in groups of 3-4  Perform 50 trials Keep a tally of your outcomes and then total them (frequency) Compute the experimental probability of each outcome P(A) = (# of times A occurred) ÷ 50  Enter your probability data in Excel Create a bar graph or pie chart displaying your probabilities  Gallery walk – observe every group’s graph

Conditions for a “fair game” a game is fair if…  all players have an equal chance of winning  each player can expect to win or lose the same number of times in the long run  each player's expected payoff is zero

Important vocabulary Trial: one repetition of an experiment e.g., flip a coin; roll a die; flip a coin and spin a spinner Random variable: a variable whose value corresponds to the outcome of a random event

More Vocabulary Expected value: the value to which the average of a random variable’s values tends after many repetitions; also called the average value Event: a set of possible outcomes of an experiment Simulation: an experiment that models an actual event

Probability A measure of the likelihood of an event Based on how often a particular event occurs in comparison with the total number of trials. Probabilities derived from experiments are known as experimental probabilities.

Experimental Probability is the observed probability (relative frequency) of an event, A, in an experiment. is found using the following formula: P(A) = number of times A occurs total number of trials Note: probability is a number between 0 and 1 inclusive. It can be written as a fraction or decimal.

Simulations A simulation is an experiment that has the same probability as an actual event. Flip a fair coin  ½ Roll a fair die  1/6, 2/6, 3/6, 4/6, 5/6 Draw a card from a standard deck (52)  ½, ¼, x/13, x/52 Hold a draw  any Spin a spinner  any (realistically 12 or fewer)

Simulations Describe a simulation that models: a) A hockey player who scores on 17% of the shots he takes b) A baseball player whose batting average is c) A randomly chosen student having a birthday during the school year a) Roll a die. Let 1 represent a goal. b) Put 3 red balls and 7 blue balls in a garbage can. Drawing a red ball represents a hit. c) Roll a die. Any number other than 1 represents the student having a birthday during the school year.

MS Excel Formulas to generate random integers Random 0 or 1 (coin toss, predict gender of a baby) =ROUND(RAND()*(1),0) Random H or T (0 or 1) =IF(ROUND(RAND()*(1),0)=0,"H","T") Random integer between 1 and 5 (football kicker p. 211 #10) =ROUND(RAND()*(4)+1,0) Random integer between 1 and n =ROUND(RAND()*(n-1)+1,0) Type a formula into a cell, then copy and paste to a group of cells to simulate multiple trials e.g., 4.1 random numbers.xls4.1 random numbers.xls Press F9 instead of ENTER to generate a random number

Class activity Play “The Coffee Game” (Investigation 1) on p. 203 including Discussion Questions on p NOTE: If you do not have pennies, or want to use technology, visit the wiki at LIEFF.CA/MDM4U and use the online coin simulator to simulate tossing 5 coins at a time!

MSIP / Homework Read through Example 2 - solution 1, p. 207 Complete pp #1, 5, 8, 9-10, 12-13

Warm up Former Toronto Blue Jays third baseman Scott Rolen has a lifetime batting average. This means that the probability that he gets a hit in any at-bat is Describe a simulation to determine whether he gets a hit in the next game (4 at bats).

Solution 1. To simulate one at-bat, we need an experiment where an event has a 286/1000 (or 2/7) probability. This could be any ONE of the following:  Put numbered balls in a drum. Choose a ball. Balls from 1 to 286 represent a hit. Replace the ball.  Generate a random number from 1 to (or 1 to 7). Any number from 1 to 286 (or 1-2) represents a hit.  Roll a 7-sided die – 1 or 2 is a hit  Spin a spinner divided into 7 segments of 360°÷7 = 51.43°. Colour two green – those sections represent a hit.  Remove all cards 8 or higher from a deck (aces low). Draw a card from the cards that remain. An A or 2 represents a hit. Replace the card. 2. Repeat 3 times to simulate 3 more at-bats.

Theoretical Probability Chapter 4.2 – An Introduction to Probability Learning goals: Calculate theoretical probabilities

Gerolamo Cardano Born: 1501, Pavia, Italy Died: 1571 in Rome (on the date he predicted astrologically) Physician, inventor, mathematician, chess player, gambler Invented combination lock, Cardan shaft Published solutions to cubic and quartic equations

Games of Chance Most historians agree that the modern study of probability began with Gerolamo Cardano’s analysis of “Games of Chance” in the 1500s. /Mathematicians/Cardan.html /Mathematicians/Cardan.html

A few terms… simple event: an event that consists of exactly one outcome (e.g., rolling a 3) sample space: the collection of all possible outcomes of an experiment (e.g., {1,2,3,4,5,6} for rolling a die) event space: the collection of all outcomes of an experiment that correspond to a particular event (e.g. {2,4,6} are the even rolls of a die)

General Definition of Probability assuming that all outcomes are equally likely, the probability of event A is: P(A)= n(A) n(S) where n(A) is the number of elements in the event space and n(S) is the number of elements in the sample space.

Example #1 When rolling a single die, what is the probability of… a) rolling a 2? b) rolling an even number? c) rolling a number less than 5? d) rolling a number greater than or equal to 5?

Example #1a When rolling a single die, what is the probability of… a) rolling a 2? A = {2}, S = {1,2,3,4,5,6} P(A) = n(A)= 1 = 0.17 n(S) 6

Example #1b When rolling a single die, what is the probability of… b) rolling an even number? A = {2,4,6}, S = {1,2,3,4,5,6} P(A) = n(A)= 3= 1 = 0.5 n(S) 6 2

Example #1c When rolling a single die, what is the probability of… c) rolling a number less than 5? A = {1,2,3,4}, S = {1,2,3,4,5,6} P(A) = n(A)= 4= 2 = 0.67 n(S) 6 3

Example #1d When rolling a single die, what is the probability of… d) rolling a number greater than or equal to 5? A = {5,6}, S = {1,2,3,4,5,6} P(A) = n(A)= 2= 1 = 0.33 n(S) 6 3

The Complement of a Set The complement of a set A, written A’ (read A complement or A prime), consists of all outcomes in the sample space that are not in the set A. If A is an event in a sample space, the probability of the complementary event, A’, is given by: P(A’) = 1 – P(A)

Example #2a When selecting a single card from a standard deck (no Jokers), what is the probability you will pick…  a) the 7 of Diamonds?  b) a Queen?  c) a face card (J, Q or K)?  d) a card that is not a face card?

Example #2a When selecting a single card from a standard deck (no Jokers), what is the probability you will pick… a) the 7 of Diamonds? P(A) = n(A) = 1 n(S) 52

Example #2b When selecting a single card from a standard deck, what is the probability you will pick… b) a Queen? P(A) = n(A) = 4 = 1 n(S) 52 13

Example #2c When selecting a single card from a standard deck, what is the probability you will pick… c) a face card (J, Q or K)? P(A) = n(A) = 12 = 3 n(S) 52 13

Example #2d When selecting a single card from a standard deck, what is the probability you will pick… d) a card that is not a face card? P(A) = n(A) = 40 = 10 n(S) 52 13

Example #2d (cont’d) Another way of looking at P(not a face card)… we know: P(face card)= 3 13 and, we know: P(A’) = 1 - P(A) So…P(not a face card) = 1 - P(face card) P(not a face card) = = 10 13

Example #2e When selecting a single card from a standard deck, what is the probability you will pick… e) a red card? P(A) = n(A) = 26 = 1 n(S) 52 2

MSIP / Homework pp # 4-7, 9, 10, 12 Next class: A look at Venn Diagrams