Welcome to MDM4U (Mathematics of Data Management, University Preparation) http://www.wordle.net/

Minds on What does it mean when you give someone a choice between 2 options and they respond with “flip a coin”?

Overall Expectations - Probability 1. Solve problems involving the probability of an event, or a combination of events, when there is a fixed number of outcomes 2. Solve problems involving the counting of ordered and unordered objects to determine the probability of an event

Introduction to Simulations and Experimental Probability Learning goals: Design a simulation for a real-world event Calculate experimental probability MSIP / Home Learning: Read through Example 2 - solution 1, p. 207 Complete pp. 209-212 #1, 5, 8-10, 12-13

Investigation – Experimental Probability How does flipping a coin relate to the gender of a baby? How likely is it that a family with 3 children has 3 boys? How could you use coins to simulate this?

Baby Simulation Decide which gender each outcome (heads and tails) will correspond to. Flip 3 coins. Record the number of heads. This represents one family with 3 children Repeat 19 more times. Calculate the experimental probability of 0, 1, 2, 3 heads. Record your results in a table. Number of Heads, X Tally Frequency P(X) II 2 2 / 20 1 IIIII II 7 7 / 20 IIIII III 8 8 / 20 3 III 3 / 20

Our results… Team 1 2 3 4 5 6 7 8 9 10 11 12 Total 32 102 82 33 32 102 82 33 Reflect: How many different outcomes are there? Do they have different probabilities?

Conditions for a “fair game” a game is fair if… all players have an equal chance of winning and equal payouts, or each player can expect to win or lose the same number of times in the long run with equal payouts, or each player's expected payoff is zero http://www.math.psu.edu/dlittle/java/probability/plinko/index.html http://probability.ca/jeff/java/utday/

A standard deck of cards

Are the following games fair? Mike and Ike… Roll a die. If a 1, 2 or 3 shows, Mike pays Ike $1. If a 4, 5 or 6 shows, Ike pays Mike $1. If an odd number shows, Mike pays Ike $1. If an even number shows, Ike pays Mike $2. Draw cards with replacement. If it’s a red card, Mike pays Ike $1. If it’s a spade, Ike pays Mike $1. Flip 3 coins. If 3 tails show, Mike pays Ike $7. Otherwise, Ike pays Mike $1.

Important vocabulary Trial: one repetition of an experiment 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 (e.g. drawing a heart) Simulation: an experiment that models an actual event (e.g. flip a coin to simulate the gender of a baby)

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 The observed probability of an event, A, in an experiment. Denoted P(A) 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, decimal or percent.

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/52, x/13 Many others if you use a partial deck Hold a draw  any Spin a spinner  any (realistically 12 or fewer)

Example 1 Describe a simulation that models: a) A hockey player who scores on 1/6 or 17% of the shots he takes takes 6 shots in a game b) A baseball player whose batting average is 0.300 gets 4 at-bats in a game c) a student in the class has a birthday during the school year a) Roll a die. Let 1 represent a goal. Roll the dice 6 times. b) Put 3 red balls and 7 blue balls in a garbage can. Drawing a red ball represents a hit. Draw 4 balls with replacement. 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.xls Press F9 instead of ENTER to generate a static random number

Recap

MSIP / Home Learning Read through Example 2 - solution 1, p. 207 Complete pp. 209-212 #1, 5, 8, 9-10, 12-13

Warm up Toronto Blue Jays shortstop Jose Reyes has a lifetime 0.286 batting average. On average he gets a hit in 2 out of every 7 at bats. Design a simulation to determine whether he gets a hit in the Blue Jays’ next game (5 at bats).

Solution 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 1 000 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 1 000 (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. Repeat 5 times to simulate 5 more at-bats.

Theoretical Probability Learning goal: Calculate theoretical probabilities Questions? pp. 209-212 #1, 5, 8-10, 12-13 MSIP / Home Learning: pp. 218-219 # 4-7, 9, 10, 12

Have you seen this store?

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. http://en.wikipedia.org/wiki/Gerolamo_Cardano http://www-gap.dcs.st-and.ac.uk/~history /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 All outcomes in the sample space that are not in the set A. Written A’ Read A-complement or A-prime n(A) + n(A’) = n(S) P(A) + P(A’) = 1 So, P(A’) = 1 – P(A)

A standard deck of cards

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? e) A red card = 26 / 52 = 1 / 2

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 = 0.02 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 = 0.08 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 = 0.23 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 = 0.77 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) = 1 - 3 = 10 13 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 / Home Learning pp. 218-219 # 4-7, 9, 10, 12 Next class: A look at Venn Diagrams