Probability and ChanceProbability and Chance

Random ExperimentRandom Experiment

An experiment is random if– The outcome depends on chance (we are not

sure of the outcome (result))– We can list all the possible outcomes in the

sample space

Example: Roll a die. We are not sure what number will be face up but we know it will be one of = {1 , 2 , 3 , 4 , 5, 6}

EventEvent

An event is a subset of the sample space

Ex : Event A is an even number – A { 2, 4, 6}

Ex: Event B is a number less than 5

- B { 1, 2, 3, 4}

Set TheorySet Theory

CompatibleCompatible

Compatible events can occur at the same time

Incompatible events can not occur at the same time

A and B can have a shared outcome A∩B ( A intersect B) is {2, 4} Event C is roll a 5 the C in incompatible with both A and B A ∩ C is empty B ∩ C is empty

Complementary EventComplementary Event

Events A and B are complementary if together they make up the sample space and there is nothing shared

A U B =

A ∩ B = 0 Example: A roll an even number B roll an odd number

Probability (P)Probability (P)

Probability is a measure of how likely it is for an event to happen.

We name a probability with a number from 0 to 1.• If an event is certain to happen, then the probability of

the event is 1. P=1• If an event is certain not to happen, then the probability

of the event is 0. P=0

ProbabilityProbabilityIf it is uncertain whether or not an event

will happen, then its probability is some fraction between 0 and 1 (part ÷ whole).

Part = # of possible favorable outcomesWhole = # of all possible outcomes

1. What is the probability that the spinner will stop on part A?

2. What is the probability that the spinner will stop on

(a) An even number?(b) An odd number?

3. What fraction names the probability that the spinner will stop in the area marked A?

ABC D

3 12

AC B

Probability QuestionsProbability Questions

Lawrence is the captain of his track team. The team is deciding on a color and all eight members wrote their choice down on equal size cards. If Lawrence picks one card at random, what is the probability that he will pick blue?

yellow

red

blue blue

blue

green black

black

Donald is rolling a number cube labeled 1 to 6. Which of the following is LEAST LIKELY?

A. an even number

B. an odd number

C. a number greater than 5

CHANCECHANCEwhat are the odds?what are the odds?

Chance is how likely it is that something will happen. To state a chance, we use a percent or a ratio ( part : part)

0 ½Probability

1

Certain not to happen

Equally likely to happen or not to happen

Certain to happen

0%50 %

50:50

Chance

100%

ChanceChance

When a meteorologist states that the chance of rain is 50%, the meteorologist is saying that it is equally likely to rain or not to rain. If the chance of rain rises to 80%, it is more likely to rain. If the chance drops to 20%, then it may rain, but it probably will not rain.

1. What is the chance of spinning a number greater than 1?

2. What is the chance of spinning a 4?

3. What is the chance that the spinner will stop on an odd number?

4. What is the chance of rolling an even number with one toss of on number cube?

1 24 3

4 12 3

5

Sample Spaces

A sample set refers to the complete set of all the possible outcomes

Example: Roll a die. What are all the possible outcomes?

Sample set “S”S = {1,2,3,4,5,6}

Sample Spaces

Example: Toss a coin. What are all the possible outcomes?

Sample space “S”S = {H, T}

Toss two coins. What is the sample space?

S = {HH, HT, TH, TT}

Events

A set of outcomes is referred to as an event.A specific outcome (part of the whole)

For example, when rolling a die the outcomes that are an even number would be referred to as an event.

Event = {2,4,6}S = {1,2,3,4,5,6}

It is clear that outcomes and events are subsets of the sample space, S.

Events

A set of outcomes is referred to as an event.A specific outcome (part of the whole)

For example, when rolling a die the outcomes that are an even number would be referred to as an event.

Event = {2,4,6}S = {1,2,3,4,5,6}

It is clear that outcomes and events are subsets of the sample space, S.

Sample Space versus Events

We use the symbol omega instead of S so that we don’t get mixed up with events

Events are given a capital letter

Ex = {1, 2, 3, 4, 5, 6}

A = { 2, 4, 6}

The sample space is all the possible outcomes of rolling a diceThe event A is rolling an even number.

Compound Events

Sometimes we are asked to find the probability of one event OR another

Sometimes we are asked to find the probability of one event AND another

What’s the difference?

Example: What is the probability of rolling a 2 OR a 4?

Compound Problems: Multiple Events What is the probability of rolling a 2 and a 4 if two die are rolled?

S = {11, 12, 13,14,15,16, 21,22,23,24,25,26, 31,32,33,34,35,36, 41,42,43,44,45,46, 51,52,53,54,55,56, 61,62,63,64,65,66}

Event {2 and 4} = {24,42}All possible outcomes = 36Possible outcomes of the stated event = 2Therefore the probability is 2 out of 36P = 0.056

Compound Events

S = {1,2,3,4,5,6}

Event {2} or {4}

There are 2 possible outcomes out of 6

P = 2/6 P = 0.33

Logical connectorsAnd , Or

When we see the probability of event A and B we multiply

When we see the probability of event A or B, we add

Example: We roll a die, what is the probability of rolling a 3 or a 5? 1/6 + 1/6 = 2/6 or 0.33Example: We roll a die and then roll it again,

what is the probability of rolling a 3 and a 5? 1/6 x 1/6 = 1/36 (much less likely)

Compound EventsIndependent versus Dependent

Events

Independent: if event A does not influence the probability of event B

Dependent: if event A does influence the probability of event B

Example: Event A: choose a marble Event B: choose a marble

They are independent if I replace the marble, dependent if I do not replace the marble

Compound EventsIndependent versus Dependent

Events

Example: there are 100 skittles20 red20 orange 20 green20 purple20 yellowWhat is the probability of choosing a red one, eating it and then choosing a yellow one?

Are these events dependent or independent?

Compound EventsIndependent versus Dependent

Events

Example: there are 100 skittles20 red20 orange 20 green20 purple20 yellowWhat is the probability of choosing a red one, eating it and then choosing a yellow one?

P(A) X P(B) = 20/100 X 20/99 (remember, I ate one)

Compound EventsIndependent versus Dependent

Events

Example: there are 100 skittles20 red20 orange 20 green20 purple20 yellowWhat is the probability of choosing 2 red one (I don’t replace the first – obviously)

P(A) X P(B) = 20/100 X 19/99 (remember, I ate one)

Compound EventsIndependent versus Dependent

Events

Example: there are 100 skittles20 red20 orange 20 green20 purple20 yellow

What is the probability of eating 1 orange, 1 green, 1 purple and then 1 green?

20/100 x 20/99 x 20/98 x 19/97

Get it?

Compound EventsIndependent versus Dependent

Events

Example: there are 100 skittles20 red20 orange 20 green20 purple20 yellow

Create your own question…