• A casino claims that its roulette wheel is truly random.
What should that mean?
• Black has come up as the winner 4 times in a row, are
chances better for betting on black or red the next time,
why?
• There are 37 spaces on the wheel, what is the probability
that the ball will land on
• Zero?
• A number divisible by three?
• A black even number?
Roulette Warm-up
• Clear desks for Ch 13 Quiz.
• After we will be taking notes on Ch 14 so have notebook
paper ready.
• Don’t forget – Helicopter Projects are due next class,
whether you are here or absent!!
WARM- up
• Today we will review the vocabulary of Ch 14,
then learn how to calculate compound probability
of disjoint and independent events.
Objective
• the long-run relative frequency of repeated
independent events gets closer and closer to the
true relative frequency as the number of trials
increases.
• For example, consider flipping a fair coin.
Law of Large Numbers
(LLN)
• Calculate the probability of six compound
probabilities; three disjoint and three
independent.
Closing Product
• I flip a fair coin one time. What is the likelihood
that it will be heads?
• I flip a fair coin 10 times. It is tails every time.
On the 11th flip what is the likelihood that it will
come up tails again?
• If I flip a coin 1000 times how many times will it
come up heads and how many times will it come
up tails?
LLN
• Commercial airplanes have an excellent
safety record, however, crashes still occur.
In the weeks following a fatal crash,
people are afraid of flying.
• A Travel agent tries to calm a traveler’s fear by
saying it’s the safest time to fly because it is
unlikely two crashes will occur so closely
together.
• How about the converse?
Example
• The proportion of times an outcome
occurs in a very long series of
repetitions.
• P(A) = # of times event A occurs
total number of occurrences
Probability
1. Two requirements for a probability:
 A probability is a number between 0 and 1.
Therefore,
For any event A, 0 ≤ P(A) ≤ 1.
Rules of Probability
2. “Something has to happen rule”:
• The probability of the set of all possible
outcomes of a trial must be 1.
• P(S) = 1 (S represents the set of all possible
outcomes.)
Rules of Probability
Red
Yellow
Green
Blue
1
0.25
0.25
0.25
0.25
2
0.5
0.3
0.2
0.10
3
.20
.4
.5
.3
4
0
0
1
0
5
.1
.2
1.2
-1.5
SPINNER EXAMPLE
3. Complement Rule:
• The set of outcomes that are not in the
event A is called the complement of A,
denoted AC.
• The probability of an event occurring is 1
minus the probability that it doesn’t occur:
P(A) = 1 – P(AC)
Rules of Probability
• A bag of marbles has 3 yellow, 4 blue and
5 green.
• What is the probability of not blue?
• What is the probability of not yellow or green?
• What is another way to write this?
EXAMPLE
• Events in which the outcome of one trial
do not influence the outcome of the other.
• Ex. Drawing a card from a deck, flipping a
coin, etc.
Independent Events
• The probability that two or more independent events will
occur.
• Used for “and” problems
P ( A  B )  P ( A) xP ( B )
• Read, “the union of A and B”
Multiplication Rule
• For example, when tossing a fair coin
twice, the probability of getting a 'Head' on
the first and then getting a 'Tail' on the
second is
• P(H and T) = P(H) × P(T)
• P(H and T) = 0.5 × 0.5
• P(H and T) = 0.25
Example
• A bag of marbles has 3 yellow, 4 blue and 5
green. Every time you draw a marble you must
replace it before drawing another one.
• What is the probability of drawing a yellow then a
green?
• What is the probability 3 blues in a row?
• What is the probability of drawing a yellow, a blue and
a green?
EXAMPLE
• Events that cannot occur at the same time.
• Flipping a single coin and getting a head and a
tail.
• Drawing a 9 and even card from a deck of
cards.
Disjoint (mutually exclusive)
• For two disjoint events A and B, the
probability that one or the other occurs is
the sum of the probabilities of the two
events.
• P(A or B) = P(A) + P(B), provided that A
and B are disjoint.
Addition Rule
• A bag of marbles has 3 yellow, 4 blue and 5
green. Every time you draw a marble you must
replace it before drawing another one.
• What is the probability of drawing a yellow or a green?
• What is the probability blue or a green?
EXAMPLE
• P(at least A) = 1- P(no A)
At least problems
• M& M’s reportedly
make 20% of the
candies yellow, 20 %
red, orange, blue and
green each make up
10% and the rest are
brown.
Example
If you pick three in a row,
what is the probability that…
1. They are all brown
2. The third one is the
first one that is
red?
3. None are yellow?
4. At least one is
green?