Download Example (cont.)

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