Transcript P(A`) = 1

PROBABILITY
Counting methods can be used to find the
number of possible ways to choose objects
with and without regard to order.
The Fundamental Counting Principle: If there
are “a” ways for one activity to occur, and
“b” ways for a second activity to occur, then
there are a  b ways for both to occur.
Example: How many different ice-cream
sundaes can be made if you have four flavors
of ice-cream, three different toppings,
and two different sauces?
4 x 3 x 2 = 24 different sundaes
A PERMUTATION is an arrangement of objects
in a specific order.
The order of the arrangement is important!
The notation for a permutation:
nPr
Where n is the total number of objects and r is
the number of objects chosen
Example: How many different ways can you
arrange five books on a shelf if you have seven
books from which to choose?
n = 7 and r = 5

7P5
= 2520 ways
FACTORIAL
If n = r, then the permutation is considered
to be a “factorial”. The formula nPr = n!
Example: How many different ways can you
arrange all six books on a shelf?
Let n = 6  n! = 6!
6 x 5 x 4 x 3 x 2 x 1 = 6!
or 720 different ways
A COMBINATION is an arrangement of objects
without regard to order.
The order of objects is not important.
The notation for a combination: nCr
Where n is the total number of objects and r
is the number of objects chosen
Example: You have twenty students in your
homeroom. How many ways can you choose
three students to represent your homeroom for
student council?
n = 20, r = 3 (the order does not matter)
20C3
= 1140 different ways
TERMINOLOGY
An outcome is the result of a single trial,
such as tossing a die.
The sample space is all the possible
outcomes.
An event is any outcome or group of
outcomes. The outcomes that match a
given event are favorable outcomes.
Event:
rolling an even number
Sample Space:
{1, 2, 3, 4, 5, 6}
Favorable Outcomes:
{2, 4, 6}
There are two kinds of probability:
Theoretical and Experimental
(Empirical)
Although both of these methods for
determining the probability of an event involve
the concept of "chance", the solutions are
obtained in two very different ways.
EXPERIMENTAL PROBABILITY
is the likelihood a specific outcome will
occur, based on repeated testing and
observation of results as in an
experiment. It is an estimate that the
event will happen based on how often the
event occurs after collecting data or
running an experiment with a large
number of trials.
It is the ratio of the number of times
an event occurs to the number of trials.
Experimental probability is calculated
after the event has occurred. By
observing the pattern of events and
how often a certain outcome has
occurred, we try to estimate how
often we can expect to see a certain
outcome in the future. The more
data that can be analyzed, the more
accurate your estimate is likely to
be. In other words, the experiment
should have many trials.
Example: To determine the
experimental probability of
tossing “tails” on a coin, toss
the coin about 100 times. Then
divide the number of times you
tossed “tails” by 100. Your
probability may not be ½ as it
would be with theoretical
probability.
Experimental Probability
On a production line, 45 toasters were found
to be defective out of the 1500 randomly
selected toasters that were tested. What is
the probability that a toaster is defective?
P(defective) =
45/
1500
= 3/100
THEORETICAL PROBABILITY
is the likelihood a specific outcome
will occur, determined by calculating
expected outcomes under ideal
circumstances.
It is the ratio of the number of
favorable outcomes to the number
of possible outcomes.
Theoretical probability is
calculated before any event
has taken place.
Example: The probability of rolling
a “5” on a die is 1/6.
Theoretical Probability
A jar contains 7 red marbles, 3
green marbles, 6 yellow marbles,
and 4 blue marbles. If a single
marble is chosen from the jar,
what is the probability it will be
red? Green? Yellow? Blue?
P(red) = 7/20
P(green) = 3/20
P(yellow) = 6/20 = 3/10
P(blue) = 4/20 = 1/5
COMPLEMENT OF AN EVENT
. The complement of an event is its “opposite,” or the
event NOT happening. If you state an event with the
symbol A, the complement of the event is written as a
A or A’. The sum of the probabilities of an event and its
complement is always equal to one.
P(A) + P(A’) = 1
The probability of the complement of an event is equal to
one minus the probability of the event:
P(A’) = 1 - P(A)
Example:
If you select one card at random from a
standard deck of cards, what is the probability
that the card is not a queen?
P(queen) = 4/52 or 1/13
 P(not a queen) = 1 – 1/13
13/
13
– 1/13
12/
13
P(Q) = 1/13
P(Q’) = 1 – 1/13 =
12/
13
The probability of an event A, is a number between 0 and 1,
inclusive.
If P(A) > P(B) then event A is more likely to occur than event B
If P(A) = P(B) then events A and B are equally likely to occur
If event A is impossible, then P(A) = 0
If event A is certain, then P(A) = 1
The complement of event A is A’:
P(A’) = 1 – P(A)
A jar of marbles has 5 green, 3 pink, 6 blue, and 4
brown. What is the probability that you select a
marble at random and it is purple?
P(purple) = 0 (an impossible event)
A bag contains certain numbers written on pieces of
paper. The numbers are {0, 2, 4, 6, 8, 10, 12}. If
you randomly pick a number from the bag, what is the
probability that the number is an even number?
P(even) = 1 (a certain event)
COMPOUND EVENTS
Compound events or combined events involve putting
together two or more events. The probability of two
compound events is found by adding the probability
of each event.
When two events cannot happen at the same time,
they are mutually exclusive events. The probability is
determined by adding the probabilities of each
event.
Mutually Exclusive Events
P(A or B) = P(A) + P(B)
Mutually Exclusive
Example:
A jar contains 1 red, 3 green, 2 blue,
and 4 yellow marbles. If a single marble
is chosen at random from the jar, what
is the probability that the marble is
yellow or green?
P(yellow or green) = P(yellow) + P(green)
= 4/10
= 7/10
+ 3/10
When two events can occur at the same
time, they are inclusive or overlapping. For
any two events which are not mutually
exclusive, the probability that an outcome
will be in one event or the other event is the
sum of their individual probabilities minus the
probability of the outcome being in both
events.
Inclusive Events
(Events that CAN happen at the same time)
P(A or B) = P(A) + P(B) - P(A and B)
Example:
Find the probability of selecting a queen or a red card
from a standard deck of cards. These events are
inclusive events because you can select two cards that
are both red and a queen: queen of hearts and queen
of diamonds.
P(queen) = 4/52
P(red) = 26/52
P(queen and red) = 2/52
P(queen or red) = P(queen) + P(red) - P(queen and red)
= (4/52 +
= 28/52
= 7/13
26/ )
52
– 2/52
INDEPENDENT EVENTS
Two events are said to be independent if
the result of the second event is not
affected by the result of the first event.
Probability of Two Independent Events
P(A and B) = P(A)  P(B)
Key Words: replaced, returned, put back,
with replacement
Example:
What is the probability that you pick a
king from a deck of cards, replace the
card, then pick another king?
P(king and king) = 4/52  4/52 = 1/13  1/13 = 1/169
DEPENDENT EVENTS
Two events are dependent events if the
occurrence of one event affects the
probability of the second event.
Probability of Two Independent Events
P(A)  P(B after A)
Key Words: without replacing, keep it, do
not return, without replacement
Example:
What is the probability that you pick a king from a
deck of cards, don’t replace the card, then pick
another king?
P(king then king) = P(king)  P(king after the 1st king)
= 4/52  3/51
= 1/13  1/17
=
1/
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