ProbabilityProbabilityCST 5CST 5

Random ExperimentRandom Experiment An experiment is considered random An experiment is considered random

when none of its possible outcomes when none of its possible outcomes can be predicted with absolute can be predicted with absolute certainty.certainty.

E.g. Rolling a die – you cannot say E.g. Rolling a die – you cannot say for sure what it will be, 1,2,3,4,5 or for sure what it will be, 1,2,3,4,5 or 6.6.

ProbabilityProbability The probability of an outcome is a The probability of an outcome is a

measure of the possibility of this measure of the possibility of this outcome occurring as the result of an outcome occurring as the result of an experiment.experiment.

The sample space of a random The sample space of a random experiment is the set of all possible experiment is the set of all possible outcomes of the experiment. outcomes of the experiment.

The symbol for sample space is The symbol for sample space is Ω -Ω -Omega.Omega.

Ω = 1,2,3,4,5,6 for the roll of a die.Ω = 1,2,3,4,5,6 for the roll of a die.

Theoretical ProbabilityTheoretical Probability The theoretical probability of an outcome The theoretical probability of an outcome

is found by reason.is found by reason. Probability of an event A occurring = P(A)Probability of an event A occurring = P(A) P(A) = P(A) = Number of desired outcomesNumber of desired outcomes Total number of all outcomes P(1) = Probability of rolling a 1 on one die = 1

6

Experimental ProbabilityExperimental Probability The experimental or empirical The experimental or empirical

probability is the relative frequency probability is the relative frequency of an outcome occurring based on of an outcome occurring based on actual results.actual results.

P(A) = P(A) = Number of desired outcomesNumber of desired outcomes Total number of trials done

Effect of ChanceEffect of Chance As the number of repetitions of an As the number of repetitions of an

random experiment increase, the random experiment increase, the role of chance decreases.role of chance decreases.

The experimental probability gets The experimental probability gets closer to the theoretical probability.closer to the theoretical probability.

E.g. Your family might have 3 boys, E.g. Your family might have 3 boys, but the ratio of boys to girls but the ratio of boys to girls worldwide is 1:1.worldwide is 1:1.

Range of ProbabilitiesRange of Probabilities Probabilities can be expressed as Probabilities can be expressed as

percentages, decimals or fractions.percentages, decimals or fractions. Probabilities range from 0 Probabilities range from 0

(impossible) to 1 (complete (impossible) to 1 (complete metaphysical certainty).metaphysical certainty).

E.g. P(raining today) = 0.5E.g. P(raining today) = 0.5 E.g. P(raining actual cats and dogs) E.g. P(raining actual cats and dogs)

= 0= 0

Odds Bodkins!Odds Bodkins! Odds are not the same as probability.Odds are not the same as probability. ““Odds for” = Odds for” = Number of successes Number of failures E.g. Odds for rolling a 1 = 1:5 “Odds against” = Number of failures Number of successes E.g. Odds against rolling a 1 = 5:1

Betting $$$Betting $$$ When betting, the ratio between the When betting, the ratio between the

amount bet and the amount to be amount bet and the amount to be won is the same as the “odds for”or won is the same as the “odds for”or “odds against.”“odds against.”

E.g. Odds for rolling a one = 1:5E.g. Odds for rolling a one = 1:5 Therefore if you bet $2, and roll a 1,Therefore if you bet $2, and roll a 1,You win 5 x $2 = $10. You win your $2 You win 5 x $2 = $10. You win your $2

back plus $8.back plus $8.

Probability of AreasProbability of Areas For games like darts or roulette, the For games like darts or roulette, the

probability of an event occurring probability of an event occurring depends on the physical size, area, or depends on the physical size, area, or volume of the desired target – volume of the desired target – compared to the overall physical size, compared to the overall physical size, area or volume.area or volume.

E.g. The chance of hitting a bull’s eye E.g. The chance of hitting a bull’s eye on a dart board depends, in part, on on a dart board depends, in part, on the area of the bull’s eye compared to the area of the bull’s eye compared to the area of the dart board.the area of the dart board.

Compound OutcomesCompound Outcomes A simple random experiment has a A simple random experiment has a

single step.single step. E.g. Rolling 1 die.E.g. Rolling 1 die. A compound random experiment has A compound random experiment has

more than 1 step.more than 1 step. E.g. Rolling 2 diceE.g. Rolling 2 dice The overall outcome depends on the The overall outcome depends on the

2 outcomes.2 outcomes.

Compound ProbabilityCompound Probability A tree diagram shows the events.A tree diagram shows the events. Compound probabilities are Compound probabilities are

multiplied across.multiplied across. Total Total probabilities are probabilities are added vertically.added vertically.

Probability of an EventProbability of an Event If the probability of an event A If the probability of an event A

occurring is P(A), then the probability occurring is P(A), then the probability of an event NOT occurring is P(A’).of an event NOT occurring is P(A’).

P(A’) = 1 – P(A)P(A’) = 1 – P(A) E.g. The probability of it raining is E.g. The probability of it raining is

0.4.0.4. Therefore the probability of it NOT Therefore the probability of it NOT

raining is 1 – 0.4 = 0.6raining is 1 – 0.4 = 0.6

Independent EventsIndependent Events Two events that can occur Two events that can occur

simultaneously (at the same time) are simultaneously (at the same time) are called independent.called independent.

E.g. A polygon can be red and have 4 E.g. A polygon can be red and have 4 sides.sides.

Two events that CANNOT occur Two events that CANNOT occur simultaneously are called Mutually simultaneously are called Mutually Exclusive.Exclusive.

E.g. You cannot flip a coin and get a E.g. You cannot flip a coin and get a heads and a tails.heads and a tails.

Conditional ProbabilityConditional Probability Conditional Probability is the probability of Conditional Probability is the probability of

an Event B occurring given that Event A an Event B occurring given that Event A has already occurred.has already occurred.

E.g. The probability of picking a student E.g. The probability of picking a student that is 16 years old and passing the course that is 16 years old and passing the course depends on the number of 16 year olds in depends on the number of 16 year olds in the class and the number of students the class and the number of students passing the course.passing the course.

Venn Diagrams and Contingency Tables Venn Diagrams and Contingency Tables are used to describe this.are used to describe this.

Contingency TablesContingency Tables Contingency tables are just a way of Contingency tables are just a way of

organizing the data.organizing the data. Note that the vertical and horizontal Note that the vertical and horizontal

totals add up to the same total in the totals add up to the same total in the bottom right corner.bottom right corner.

The numbers arrived at can be used The numbers arrived at can be used to calculate conditional probability to calculate conditional probability problems.problems.

Venn DiagramsVenn Diagrams Venn Diagrams are used to visually show Venn Diagrams are used to visually show

how information is grouped in two different how information is grouped in two different ways.ways.

The The Ω symbol stands for the set of all Ω symbol stands for the set of all outcomes.outcomes.

All the data should appear inside the box.All the data should appear inside the box. Some of the data will be in one group or both Some of the data will be in one group or both

groups.groups. If it is in both groups, it occurs where the two If it is in both groups, it occurs where the two

circles intersect.circles intersect.

Venn DiagramsVenn Diagrams A U B (read A Union B) is the set of aa A U B (read A Union B) is the set of aa

data that is in Group A and Group Bdata that is in Group A and Group B A A ∏ B (read A intersect B) is the set of ∏ B (read A intersect B) is the set of

data that is in the intersection of data that is in the intersection of groups A and B.groups A and B.

PPAB is the probability of choosing an B is the probability of choosing an outcome in Group B, given that it is in outcome in Group B, given that it is in Group A.Group A.

SituationSituation The residents of a Montreal The residents of a Montreal

neighbourhood were asked which of the neighbourhood were asked which of the 3 daily newspapers they prefer.3 daily newspapers they prefer.

Of the survey’s 190 respondents, 98 said Of the survey’s 190 respondents, 98 said they preferred the Montreal Gazette.they preferred the Montreal Gazette.

72 said they liked the Toronto Globe & 72 said they liked the Toronto Globe & Mail.Mail.

Of the women, 40 prefer the G&M, 38 Of the women, 40 prefer the G&M, 38 the Gazette, and 15 the National Post.the Gazette, and 15 the National Post.

Contingency TableContingency TableNewsNewsSexSex

G&MG&M GazetteGazette Nat.Nat.PostPost

TotalTotal

MaleMale

FemaleFemale 4040 3838 1515

TotalTotal 7272 9898 190190

P(reads G&M, given a male)P(reads G&M, given a male)

G&M (72)Male (97) 65 4032

Of the 97 men, 32 read the G&M. The probability is 32/97 or 0.330

P(reads Gaz, given a P(reads Gaz, given a female)female)

Gazette (98)Female 93 55 6038

Of the 93 women 38 read the Gaz. The probability is 38/93 or 0.409

P(is male, given reads G&M)P(is male, given reads G&M)

Male(97)G&M (72) 40 6532

Of the 72 that read the G&M, 32 read the G&M. The probability is 32/72or 0.440

P(is female, given reads the Nat P(is female, given reads the Nat Post)Post)

Female (93)Nat Post (20) 5 7815

Of the 20 that read the Nat Post, 15 are female. The probability is 15/20 or 0.75.

ActivityActivity Page 30, Q. 1 -5Page 30, Q. 1 -5

Expected ValueExpected Value The expected value is the sum of the The expected value is the sum of the

products and their probabilities.products and their probabilities. It indicates the relative chance of winning It indicates the relative chance of winning

or losing at the game.or losing at the game. Expected Value = pExpected Value = p11oo11 + p + p22oo22 + p + p33oo33…… Where p is the probability and o is the net Where p is the probability and o is the net

gain or net loss.gain or net loss. Net Gain means win – bet if the bet is not Net Gain means win – bet if the bet is not

returned, or just the win if the bet is returned, or just the win if the bet is returned. Money lost means the bet lost.returned. Money lost means the bet lost.

SituationSituation E.g. A game involves paying $2 and rolling a E.g. A game involves paying $2 and rolling a

die. If a ‘3’ is rolled, you win $5 plus your bet, die. If a ‘3’ is rolled, you win $5 plus your bet, otherwise you lose.otherwise you lose.

Is this game fair? Would you play this game?Is this game fair? Would you play this game? Determine the Expected ValueDetermine the Expected Value

E.V. = pE.V. = p11oo11 (net gain) +p (net gain) +p22oo2 2 (loss)(loss) = 1/6 (5) + 5/6 (-2)= 1/6 (5) + 5/6 (-2) = 5/6 -10/6= 5/6 -10/6 = -5/6= -5/6 =-0.833 Therefore NOT FAIR!!! Keep your $=-0.833 Therefore NOT FAIR!!! Keep your $

SituationSituation A game involves a 52-card deck. If a A game involves a 52-card deck. If a

card is picked is a Spade, you pay card is picked is a Spade, you pay $20.$20.If the card picked is a Heart, you win If the card picked is a Heart, you win $8. If the card picked is a Diamond, $8. If the card picked is a Diamond, you win $4.you win $4.

If the game is fair, and generates $45 If the game is fair, and generates $45 in profits, then how much money in profits, then how much money should a club cost you?should a club cost you?

SituationSituation The game is fair, so the E.V. is 0.The game is fair, so the E.V. is 0. E.V. = pE.V. = p11oo11 (win) +p (win) +p22oo2 2 (lose)(lose) 0 = ¼ (-20) + ¼ (8) +1/4 (4) + ¼(x)0 = ¼ (-20) + ¼ (8) +1/4 (4) + ¼(x) SolveSolve 0 = -5 + 2 + 1 +0.25x0 = -5 + 2 + 1 +0.25x 2 2 = = 0.25x0.25x 0.25 0.250.25 0.25 8 = x…….so a club should win $88 = x…….so a club should win $8

What is Fair for You?What is Fair for You? Rule 1: Play only those games in Rule 1: Play only those games in

which the mathematical expectation is which the mathematical expectation is positive or at least zero.positive or at least zero.

Rule 2: If given a choice between 2 Rule 2: If given a choice between 2 games, choose the one with the games, choose the one with the greater mathematical expectation.greater mathematical expectation.

This works over the long term; chance This works over the long term; chance plays a greater role in the short term.plays a greater role in the short term.

ActivityActivity Do Page 39, Q. 1-9Do Page 39, Q. 1-9 Do Page 62, Q. 1-9Do Page 62, Q. 1-9 Do Page 63, Q. 10-18Do Page 63, Q. 10-18 Time for an Evaluation of your Time for an Evaluation of your

CompetenciesCompetencies