Download Example (cont.)

Warm-Up
1. 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?
Chance
The Mean and Standard Deviation of
a Probability Model
•
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 x¯ of the observed values gets closer and closer to μ.
(This is true for trials with numerical outcomes and a finite mean μ.)
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 Examples
Coin 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 Diagrams
Illustrate 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-Up
1. 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-Up
1. 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 head and a 4)
P(a head and an odd #)
P(a tail and a # > 1)
P(a tail and a # < 2)
P(A)
P(B)
P(A and B)
Homework Check
1. A coin is tossed three times. Find the
probability that the result is at least two
heads.
A. 1/2
B. 1/3
B. 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. 1/8
C. 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. 9/16
C. 15/128
D. 5/16
4. Two events that have nothing in common are
called:
A. inconsistent
B. mutually exclusive
C. 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. 20/72
C. 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. 0.6
B. 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 then
P(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.
Z
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
1 2 3 4 5 6 7 8 9
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
Is it a fair game?
A spinner has 5 equal sectors numbers 1 to 5, it is spun many times
5
If the spinner stops on an even number
1
team A gets 3 points
4
If the spinner stops on an odd number
team B gets 2 points
2
3
Warm Up
1. 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
•
A  B denotes the union of the sets A and B.
– A or B or both A and B.
•
A  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  B   , where  represents the empty set.
– A and B are said to be mutually exclusive.
Examples (cont.)
• If A is the set of all factors of 36 and B is the set of all
factors of 54, find:
–AUB
–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