Warm-Up1. What is Benford’s Law?2. What two numbers do all probabilities fall

between?3. What does equally likely outcomes mean?4. What is the formula for equally likely

outcomes?5. What distribution does Benford’s Law fall

under?6. What distribution does equally likely

outcomes follow?

Probability: The Mathematics of Chance The Mean and Standard Deviation of a Probability Model

2

• Mean of a Continuous Probability Model– Suppose the area under a density curve was cut out of solid

material. The mean is the point at which the shape would balance.

Law of Large Numbers As a random phenomenon is repeated a large number of times:

The proportion of trials on which each outcome occurs gets closer and closer to the probability of that outcome, and

The mean ¯ of the observed values gets closer and closer to μ. (This is true for trials with numerical outcomes and a finite mean μ.)

x

Experimental Probability• Observing the results of an experiment

• An event which has a 0% chance of happening (i.e. impossible) is assigned a probability of 0.

• An event which has a 100% chance of happening (i.e. is certain) is assigned a probability of 1.

• All other events can then be assigned a probability between 0 and 1.

Experimental Probability Terminology • Number of Trials – the total number of times the

experiment is repeated.• The outcomes – the different results possible for

one trial of the experiment.• Frequency – the number of times that a particular

outcome is observed.• Relative Frequency – the frequency of an outcome

expressed as a fraction or percentage of the total number of trials. – **experimental probability = relative frequency**

Sample Space• The set of all possible outcomes of an experiment

• Examples:– Tossing a coin

– Rolling a die

Example

• We roll two dice and record the up-faces in order (first die, second die)

– What is the sample space S?

– What is the event A: “ roll a 5”?

Probability Model• Example: Rolling two dice– We roll two dice and record the up-faces in order (first die,

second die)– All possible outcomes• (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)• (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)• (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)• (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)• (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)• (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

– “Roll a 5” : {(1,4) (2,3) (3,2) (4,1)}

Probability Models

• Give me the sample space for:Flipping two coins.

Experimental Probability ExamplesCoin Tossing & Dice Rolling

Coin Toss

Dice

2-D Grids:1. Illustrate the possible outcomes when 2

coins are tossed.

2. Illustrate the possible outcomes for the sum of 2 dice being rolled.

2-D Grid

2-D Grids:

3. Illustrate the possible outcomes when tossing a coin and rolling a die.

Tree DiagramsIllustrate the possible outcomes when

• tossing 2 coins• drawing 2 marbles from a bag containing red,

green and yellow marbles

Tree Diagrams

Theoretical Probability• For fair spinners, coins or die (where a particular

outcome is not weighted) the outcomes are considered to have an equal likelihood.

• For a fair dice, the likelihood of rolling a 3 is the same as rolling a 5… both 1 out of 6

• This is a mathematical (or theoretical) probability and is based on what we expect to occur.– A measure of the chance of that event occurring in any

trial of the experiment

Warm-Up1. Have your homework out on your desk.2. Create a tree diagram for the following.- Flipping two coins- Pulling a marble out of a bag full of blue,

green and yellow marbles.3. How many outcomes total?4. What is the probability of pulling a blue

marble out of the bag?5. What is the probability of flipping heads in

the scenario?

Homework Answers

Theoretical Probability Examples• A ticket is randomly selected from a basket

containing 3 green, 4 yellow and 5 blue tickets. Determine the probability of getting:– A green ticket– A green or yellow ticket– An orange ticket– A green, yellow or blue ticket

Complementary Events• An ordinary 6-sided die is rolled once. Determine

the chance of:– Getting a 6– Not getting a 6– Getting a 1 or 2– Not getting a 1 or 2

Warm-Up1. Have your homework out on your desk.2.

Homework Check

More Grids to Find Probabilities• Use a two-dimensional grid to illustrate the sample

space for tossing a coin and rolling a die simultaneously. From this grid determine the probability of:– Tossing a head– Getting a tail and a 5– Getting tail or a 5

More Grids to Find Probabilities (cont.)• 2 circular spinners, each with 1 – 10 on their edges are

twirled simultaneously. Draw a 2D grid of the possible outcomes and use your grid to determine the probability of getting– A 3 with each spinner– A 3 and a 1– An even result for each spinner

Spinner

Warm-Up Compound Events• Create a 2-D grid for the following situation. • A coin is tossed and at the same time, a die

is rolled. The result for the coin will be outcome A and the die, outcome B.

P(A and B) P(A) P(B) P(A and B)

P(a head and a 4)

P(a head and an odd #)

P(a tail and a # > 1)

P(a tail and a # < 2)

Homework Check

1. A coin is tossed three times. Find the probability that the result is at least two heads.

A. 1/2 B. 1/3B. C. 3/8 D. None of these

2. A card is drawn from a standard deck of 52 cards. Then, a second card is drawn from the deck (without replacing the first one). Find the probability that a red card is selected first and a spade is selected second.

A. 1/3 B. 1/8C. 13/102 D. None of these

3. From an urn containing 16 cubes of which 5 are red, 5 are white, and 6 are black, a cube is drawn at random. Find the probability that the cube is red or black.

A. 11/16 B. 9/16C. 15/128 D. 5/16

4. Two events that have nothing in common are called:A. inconsistent B. mutually exclusiveC. complements D. Both A and B

5. A bag contains 5 white balls and 4 red balls. Two balls are selected in such a way that the first ball drawn is not replaced before the next ball is drawn. Find the probability of selecting exactly one white ball.

A. 12/72 B. 20/72C. 5/9 D. 4/5

6. A and B are two events such that p(A) = 0.2 and p(B) = 0.4. If , find .

A. 0.45 B. 0.6B. C. 0.85 D. None of these

Warm-Up

1. Create a tree diagram. When you go to a restaurant you have a choice for three course meals your 4 salad choices, 6 entrees, and 5 dessert choices.

2. How many possible outcomes are there?

Independent Events• Events where the occurrence of one of the events does not

affect the occurrence of the other event.

• In general, if A and B are independent events, then – P(A and B) = P(A) x P(B)

• Ex: a coin and a die are tossed simultaneously. Determine the probability of getting a head and a 3 without using a grid.

Using Tree Diagrams

Examples:

Examples (cont.):• Carson is not having much luck lately. His car will only start 80% of

the time and his moped will only start 60% of the time.

– Draw a tree diagram to illustrate the situation.• 1st set of branches for the car, 2nd set of branches for the

moped

– Use the diagram to determine the chance that• Both will start• He has to take his car.• He has to take the bus.

Dependent Events• Think About It: A hat contains 5 red and 3 blue tickets. One ticket is

randomly chosen and thrown out. A second ticket is randomly selected. What is the chance that it is red?

• Not independent; the occurrence of one of the events affects the occurrence of the other event.

• If A and B are dependent events thenP(A then B) = P(A) x P(B given that A has occurred)

Examples:• A box contains 4 red and 2 yellow tickets. Two tickets are randomly

selected one by one from the box, without replacement. Find the probability that:

– Both are red– The first is red and the second is yellow

Examples (cont.):• A hat contains tickets with numbers 1 – 20 printed on them. If 3

tickets were drawn from the hat without replacement, determine the probability that all are prime numbers.

Examples (cont.):• A box contains 3 red, 2 blue and 1 yellow marble. Draw a tree

diagram to represent drawing 2 marbles.• With replacement Without replacement

• Find the probability of getting two different colors:– If replacement occurs– If replacement does not occur

Examples (cont.):• A bag contains 5 red and 3 blue marbles. Two marbles are

drawn simultaneously from the bad. Determine the probability that at least one is red.

Is it a fair game?

Questions will be put up on the board

For each, you have to decide if the game is:

Fair

Not Fair

Once you decide on your answer, write it on your mini-whiteboard

Only show the you answer when asked

Z

Is it a fair game?

Three people have boards like the one shown below. You throw a coin onto

a board, if it lands on a shaded square you win

(assume the coin lands exactly in a square)

Z

Is it a fair game?

A marble is picked from the container by the teacher

If its red the girls get a point, if its blue the boys get a point

Z

Is it a fair game?

Nine cards numbers 1 to 9 are used for a game

A card is drawn at random

If a multiple of 3 is drawn team A gets a point

If a square number is drawn team B gets a point

If any other number is drawn team C gets a point

Z

1 2 3 4 65 7 8 9

Is it a fair game?

A spinner has 5 equal sectors numbers 1 to 5, it is spun many times

If the spinner stops on an even number

team A gets 3 points

If the spinner stops on an odd number

team B gets 2 points

Z

5 1

2

3

4

Warm Up1. Get your homework out.2. A box contains 4 red marbles, 5 blue marbles and 1

green marble. We select 2 marbles without replacement. Determine the probability of getting:– At least 1 red marble– One green and one blue marble

Sets & Venn Diagrams• A Venn diagram consists of a rectangle which

represents the sample space and at least 1 circle within it representing particular events.

Examples• The Venn diagram represents a sample space of students.

The event E, shows all those that have blue eyes. Determine the probability that a student– Has blue eyes

Examples (cont.)• Draw a Venn diagram and shade the regions to represent

the following:– 1. In A but not in B 2. Neither in A nor B

• denotes the union of the sets A and B. – A or B or both A and B.

• denotes the intersection of sets A and B. – All elements common to both sets.

• Disjoint sets do not have elements in common. So

– A and B are said to be mutually exclusive.

B A

B A

set.empty therepresents where,B A

Examples (cont.)• If A is the set of all factors of 36 and B is the set of all

factors of 54, find:– A U B– A ∩ B

Examples (cont.)

Examples (cont.)• In a class of 30 students, 19 study Physics, 17 study

Chemistry and 15 study both. Display this in a Venn diagram and find the probability that a student studies:– Both– At least 1 of the subjects– Physics, but not Chemistry– Exactly one of the subjects– Neither– Chemistry given that the student also studies physics

Warm-Up Find the following probabilities.

Laws of Probability• For 2 events A and B,– P (A U B) = P(A) + P(B) – P (A ∩ B)

• Example: P(A) = 0.6, P(A U B) = 0.7 and P(A ∩ B) = 0.3– Represent this using a Venn diagram and find P(B)

Mutually Exclusive Events• If A and B are mutually exclusive the intersection is the

empty set and equals 0.– So the law becomes:

• Example: A box of chocolate contains 6 with hard centers (H) and 12 with soft centers (S). – Are H and S mutually exclusive?– Find P(H ∩ S)– Find P(H U S)

Laws of Probability (cont.)• Conditional Probability (dependent events):– A | B represents “A occurs knowing B has occurred”

– It follows that:

Example• In a class of 40, 34 like bananas, 22 like pineapples, and 2

dislike both fruits. Find the probability that a student:– Likes both– Likes at least one– Likes bananas give that they like pineapples– Dislikes pineapples given that they like bananas

Example (cont.)• Box A contains 3 red and 2 white tickets. Box B contains 4

red and 1 white. A die with 4 faces marked A and 2 faces marked B is rolled and used to select a box. Then we draw a ticket. Find the probability that:– The ticket is red– The ticket was chosen from B given it is red.

Using Definitions• If A and B are independent, how do we find P(A and B)?

– When 2 coins are tossed, A is the event of getting 2 heads. When a die is rolled, B is the event of getting a 5 or 6. Prove that A & B are independent

Using Definitions (cont.)• If A and B are mutually exclusive, what has to be true?

– P(A) = ½ and P(B) = 1/3, find P(A U B) if:• A and B are mutually exclusive• A and B are independent