Transcript PROBABILITY
PROBABILITY
Chapter 9
Section 9-1
Review Percents and
Probability
Experiment
An activity that is used to
produce data that can be
observed and recorded
Example – rolling a die
Example - tossing a coin
Example – drawing a card
Outcome
The result of each trial of an
experiment.
Event
Any one of the possible
outcomes or combination of
possible outcomes of an
experiment
Experimental Probability
Represents an estimate of the
likelihood of an event, E, or
desired outcome
P(E)= # of observations of E
total # of observations
Theoretical Probability
P(E) = # of favorable outcomes
# of possible outcomes
Sample Space
The set of all possible
outcomes of the experiment
Tossing a coin –
S = {H, T}
Rolling a dice –
S = {1, 2, 3, 4, 5, 6}
Tree Diagram
A diagram that lists one part
of an event and then adds
branches to show all the
outcomes involving that part
of the event
Example
In an experiment, a coin is
tossed and a number cube is
rolled.
Make a tree diagram
beginning with the possible
outcomes of the coin toss
Relative Frequency
Compares the number of times
the outcome occurs to the total
number of observations
Example
The more often you toss a
coin, the closer you will
come to tossing an equal
number of heads and tails.
Section 9-2
Problem Solving Skills:
Simulations
Section 9-3
Compound Events
Compound Event
Made up of two or more
simpler events
Probability of a compound
event is the probability of
one event and/or another
occurring
Probability
The probability of a
compound event is
represented by P(A ∩ B)
The probability of one event
or another occurring is
written P(A B)
MUTUALLY EXCLUSIVE EVENTS
Events that cannot occur at
the same time
Example – A die is rolled.
The events, getting an even
number and getting an odd
number are mutually
exclusive.
MUTUALLY EXCLUSIVE EVENTS
If two events A and B are
mutually exclusive then
AB=Ø
and
Mutually Exclusive Events
For mutually exclusive
events only:
P(A B) = P(A) + P(B)
EXAMPLE – MUTUALLY EXCLUSIVE
EVENTS
• Suppose a die is tossed.
• Let A be the event that an
even number turns up
• Let B be the event that an
odd number turns up, then
Mutually Exclusive Events
A = {2, 4, 6}, and B = {1, 3, 5}
AB=Ø
THEOREM
If A and B are not mutually
exclusive events, then
P(A B)
=
P(A) + P(B) – P(A B)
Example A card is drawn at random from a
deck of 52 playing cards. Find the
probability that the card is a heart
or an ace.
A = card is a heart
B = card is an ace
P(A B) = P(A) + P(B) – P(A B)
Section 9-4
Independent and
Dependent Events
INDEPENDENT EVENTS
Two events are independent
if the result of the second
event is not affected by the
result of the first event.
INDEPENDENT EVENTS
The events A and B are
independent if, and only if
P(A B) = P(A) • P(B)
Example
A bag contains 3 red marbles, 4
green marbles and 5 blue
marbles. One marble is taken at
random and then replaced. Then
another marble is taken at
random.
Find the probability that the 1st
marble is red and the 2nd is blue.
DEPENDENT EVENTS
Two events are dependent if
the result of one event is
affected by the result of
another event
DEPENDENT EVENTS
The result of event A affects
event B
P(A B) =
P(A) • P(B, given that A occurred)
P(A) • P(B|A)
Example
A bag contains 3 red marbles, 4
green marbles and 5 blue
marbles. One marble is taken
at random and is not replaced.
Then another marble is taken at
random.
Find the probability that the 1st
marble is red and the 2nd is
blue.
Section 9-5
Permutations and
Combinations
Fundamental Counting Principle
If there are two or more
stages of an activity, the total
number of possible
outcomes is the product of
the number of possible
outcomes for each stage
Example
At a pizza place there are
three sizes (Large, Medium,
and Small). There are also
five choices of toppings
(cheese, pepperoni,
sausage, onions, peppers).
How many different pizzas
with one topping could a
customer order?
What is the probability that a
customer will order a
Medium pizza with sausage?
Example
A store sells shirts in 8 sizes.
For each size, there is a
choice of 5 colors. For each
color, there is a choice of 6
patterns. How many
different shirts does the store
have?
What is the probability that a
customer will buy a large
shirt that is blue with stripes?
PERMUTATION
An arrangement of items in
a particular order.
n! (n factorial)
n(n-1)(n-2)…(2)(1)
FACTORIAL
5! = 5 x 4 x 3 x 2 x 1
0! = 1
EXAMPLE PERMUTATIONS
How many different “ways”
can the letters a, b, and c
be arranged if all the
letters are used?
3!
(a,b,c), (a,c,b), (b,c,a),
(b,a,c), (c,a,b), (c,b,a)
PERMUTATIONS
NO REPETITIONS
Uses only a part of the set
without repetitions
nPr = n!__
(n-r)!
n = number of items
r = number of items taken at
a time
EXAMPLE PERMUTATIONS
How many different “ways”
can the letters a, b, c, and
d be arranged if only three
different letters are used?
4!__
(4 - 3)!
ANSWER
How many different “ways”
can the letters a, b, c, and
d be arranged if only three
different letters are used?
4! = 24
COMBINATION
An arrangement of items in
which order is not
important.
nCr = n!__
(n-r)!r!
COMBINATION
nCr =
n!__
(n-r)!r!
n = number of different items
r = number of items taken at
a time
EXAMPLE COMBINATIONS
How many different ways can
a 2-person committee be
chosen from 8 people if there
are no restrictions?
8!____
(8 - 2)!2!
EXAMPLE COMBINATIONS
A random drawing is held to
determine which 2 of the 6
members of the math club will
be sent to a regional math
contest.
How many different pairs of two
could be sent to the contest?
EXAMPLE COMBINATIONS
How many combinations of
three letters could you make
out of the letters a, b, c, d, e,
and f?
EXAMPLE COMBINATIONS
A popular touring band has
20 songs. How many
combinations of songs can
the band play in their
opening 3-song set?
Section 9-6
Scatter Plots and Boxplots
SCATTER PLOT
A type of visual display
showing a relationship
between two sets of data,
represented by
unconnected points on a
grid.
Factory Wages
14
12
Hourly Pay
10
8
6
4
2
0
0
2
4
6
8
Years of Experience
10
12
14
16
BOX-AND-WHISKER PLOT
A type of visual display
showing how data are
dispersed around a
median. It does not show
specific items in the data.
but
BOX-AND-WHISKER PLOT
It shows the median and the
extremes of a set of data.
The lower half of the data,
called the lower quartile, and
the median of the upper half
called the upper quartile.
END