Download Probability Summary: The probability of an event is the measure of

Probability
Summary:
The probability of an event is the measure of the chance that
the event will occur as a result of an experiment. The
probability of an event A is the number of ways event A can
occur divided by the total number of possible outcomes. The
probability of an event A, symbolized by P(A), is a number
between 0 and 1, inclusive, that measures the likelihood of
an event in the following way:


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.
Certain / Impossible
Experiment 5:
A total of five cards are chosen at random
from a standard deck of 52 playing cards.
What is the probability of choosing 5 aces?
Probability:
It is impossible to choose 5 aces since a
standard deck of cards has only 4 of a kind.
This is an impossible event.
P(5 aces) =
0
=0
52
Experiment 6: A glass jar contains 15 red marbles. If a single marble
is chosen at random from the jar, what is the
probability that it is red?
Probability:
Choosing a red marble is certain to occur since all 15
marbles in the jar are red. This is a certain event.
P(red) =
15
=1
15
Summar The probability of an event is the measure of the chance that the event will occur as a
y:
result of the experiment. The probability of an event A, symbolized by P(A), is a
number between 0 and 1, inclusive, that measures the likelihood of an event in the
following way:




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.
Sample Spaces
Unit 6
Experiment 1:
What is the probability of each
outcome when a dime is tossed?
Outcomes:
The outcomes of this experiment
are head and tail.
Probabilities:
P(head) =
1
2
P(tail)
1
2
=
Definition: The sample space of an experiment is the set of all possible
outcomes of that experiment.
The sample space of Experiment 1 is: {head, tail}
Experiment 4:
A glass 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 of each
outcome?
Sample Space:
{red, green, blue, yellow}
Probabilities:
=
1
10
P(green) =
3
10
P(blue)
=
2 1
=
10 5
P(yellow) =
4 2
=
10 5
P(red)
Complement
of an Event
Experiment 1:
Unit 6
A spinner has 4 equal sectors colored yellow, blue,
green and red. What is the probability of landing on a
sector that is not red after spinning this spinner?
Sample Space: {yellow, blue, green, red}
Probability:
The probability of each outcome in this experiment is
one fourth. The probability of landing on a sector that is
not red is the same as the probability of landing on all
the other colors except red.
P(not red) =
1 1 1 3
+ + =
4 4 4 4
In Experiment 1, landing on a sector that is not red is the complement of landing on a sector that
is red.
Definition: The complement of an event A is the set of all outcomes in the sample space
that are not included in the outcomes of event A. The complement of event A
is represented by
(read as A bar).
Rule:
Given the probability of an event, the probability of its complement can be
found by subtracting the given probability from 1.
P( ) = 1 - P(A)
You may be wondering how this rule came about. In the last lesson, we learned that the
sum of the probabilities of the distinct outcomes within a sample space is 1. For example,
the probability of each of the 4 outcomes in the sample space above is one fourth, yielding a
sum of 1. Thus, the probability that an outcome does not occur is exactly 1 minus the
probability that it does. Let's look at Experiment 1 again, using this subtraction principle.
Experiment 1:
A spinner has 4 equal sectors colored yellow,
blue, green and red. What is the probability of
landing on a sector that is not red after spinning
this spinner?
Sample Space:
{yellow, blue, green, red}
Probability:
P(not red) = 1 - P(red)
=1-
=
1
4
3
4
periment 2:
A single card is chosen at random from
a standard deck of 52 playing cards. What is the
probability of choosing a card that is not a king?
Probability:
P(not king) = 1 - P(king)
=1 -
Summary:
=
48
52
=
12
13
4
52
The probability of an event is the measure of the chance that the event will occur
as a result of the experiment. The probability of an event A, symbolized by P(A), is
a number between 0 and 1, inclusive, that measures the likelihood of an event in
the following way:





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 . P( ) = 1 - P(A)
Mutually Exclusive
Events
Experiment 1:
A single card is chosen at random
from a standard deck of 52 playing
cards. What is the probability of
choosing a 5 or a king?
Possibilities:
1. The card chosen can be a 5.
2. The card chosen can be a king.
Experiment 2:
A single card is chosen at random
from a standard deck of 52 playing
cards. What is the probability of
choosing a club or a king?
Unit
6
Possibilities:
1. The card chosen can be a club.
2. The card chosen can be a king.
3. The card chosen can be a king and
a club (i.e., the king of clubs).
In Experiment 1, the card chosen can be a five or a king, but not both at the
same time. These events are mutually exclusive. In Experiment 2, the card
chosen can be a club, or a king, or both at the same time. These events are
not mutually exclusive.
Definition:
Two events are mutually exclusive if they cannot occur at the
same time (i.e., they have no outcomes in common)
Experiment 3:
A single 6-sided die is
rolled. What is the
probability of rolling an odd
number or an even
number?
Possibilities:
1. The number rolled can
be an odd number.
2. The number rolled can
be an even number.
Events:
These events are mutually
exclusive since they
cannot occur at the same
time.
Experiment 4:
A single 6-sided die is
rolled. What is the
probability of rolling a 5 or
an odd number?
Possibilities:
1. The number rolled can be a 5.
2. The number rolled can be an odd
number (1, 3 or 5).
3. The number rolled can be a 5
and odd.
Events:
These events are not mutually exclusive since they can occur at the same time
In each of the three experiments above, the events are mutually exclusive. Let's look at
some experiments in which the events are non-mutually exclusive.
Experiment 4:
A single card is chosen at random from a standard
deck of 52 playing cards. What is the probability of
choosing a king or a club?
Probabilities:
P(king or club) = P(king) + P(club) - P(king of clubs)
=
4
52
=
16
52
=
4
13
+
13
52
-
1
52
In Experiment 4, the events are non-mutually exclusive. The addition causes the king of
clubs to be counted twice, so its probability must be subtracted. When two events are nonmutually exclusive, a different addition rule must be used.
Addition Rule 2:
When two events, A and B, are non-mutually exclusive, the probability that A
or B will occur is:
P(A or B) = P(A) + P(B) - P(A and B)
In the rule above, P(A and B) refers to the overlap of the two events. Let's apply
this rule to some other experiments
Experiment 5:
In a math class of 30 students, 17 are boys and 13
are girls. On a unit test, 4 boys and 5 girls made an A
grade. If a student is chosen at random from the
class, what is the probability of choosing a girl or an
A student?
Probabilities:
P(girl or A) = P(girl) + P(A) - P(girl and A)
Experiment 6:
=
13
30
=
17
30
9
30
+
5
30
On New Year's Eve, the
probability of a person having a
car accident is 0.09. The
probability of a person driving
while intoxicated is 0.32 and
probability of a person having a
car accident while intoxicated is
0.15. What is the probability of a
person driving while intoxicated
or having a car accident?
Probabilities:
P(intoxicated or accident) = P(intoxicated) + P(accident) - P(intoxicated and accident)
=
=
Summary:
0.32
0.26
+
0.09
-
0.15
To find the probability of event A or B, we must first determine
whether the events are mutually exclusive or non-mutually
exclusive. Then we can apply the appropriate Addition Rule:
Addition Rule 1: When two events, A and B, are
mutually exclusive, the probability
that A or B will occur is the sum of
the probability of each event.
P(A or B) = P(A) + P(B)
Addition Rule 2:: When two events, A and B, are nonmutually exclusive, there is some
overlap between these events. The
probability that A or B will occur is
the sum of the probability of each
event, minus the probability of the
overlap.
P(A or B) = P(A) + P(B) - P(A and B)
Independent
Events
Unit
6
Experiment 1: A dresser drawer contains one pair of socks with
each of the following colors: blue, brown, red,
white and black. Each pair is folded together in a
matching set. You reach into the sock drawer and
choose a pair of socks without looking. The first
pair you pull out is red --the wrong color. You
replace this pair and choose another pair of
socks. What is the probability that you will choose
the red pair of socks twice?
There are a couple of things to note about this experiment. Choosing two pairs of socks
from the same drawer is a compound event. Since the first pair was replaced, choosing a
red pair on the first try has no effect on the probability of choosing a red pair on the second
try. Therefore, these events are independent.
Definition: Two events, A and B, are independent if the fact that A occurs does not affect the
probability of B occurring.
Some other examples of independent events are:




Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
Choosing a marble from a jar AND landing on heads after tossing a coin.
Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the
second card.
Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.
To find the probability of two independent events that occur in sequence, find the probability
of each event occurring separately, and then multiply the probabilities. This multiplication
rule is defined symbolically below. Note that multiplication is represented by AND.
Multiplication Rule 1: When two events, A and B, are independent, the probability of both
occurring is:
P(A and B) = P(A) · P(B)
Experiment 1: A dresser drawer contains one pair of socks of
each of the following colors: blue, brown, red,
white and black. Each pair is folded together in
matching pairs. You reach into the sock drawer
and choose a pair of socks without looking. The
first pair you pull out is red -the wrong color. You
replace this pair and choose another pair. What is
the probability that you will choose the red pair of
socks twice?
Probabilities:
P(red)
=
1
5
P(red and red) = P(red) · P(red)
=
1
5
=
1
25
1
5
·
Experiment 2: A coin is tossed and a single 6-sided die is rolled.
Find the probability of landing on the head side of
the coin and rolling a 3 on the die.
Probabilities:
P(head)
=
1
2
P(3)
=
1
6
P(head and 3) = P(head) · P(3)
=
1
2
=
1
12
·
1
6
Experiment 3: A card is chosen at random from a deck of 52 cards. It is
then replaced and a second card is chosen. What is the probability of
choosing a jack and an eight?
P(jack)
=
4
52
P(8)
=
4
52
P(jack and 8) = P(jack) · P(8)
=
4
52
=
16
2704
=
1
169
·
4
52
Probabilities: