Download Probability

Probability – the likelihood that
an event will occur.
Probability is usually expressed as a
real number from 0 to 1.
The probability of an impossible event
is 0.
The probability of a certain event is 1.
Experimental Probability –
Probability that is determined by
collecting data from oberservations
P(event) = # of times the event occurs
# of trials
Example of Experimental Probability…
A baseball player got a hit 21 times
in 60 at-bats. Find the
experimental probability of his
getting a hit.
21 = 7
60 20
Theoretical Probability – Probability
based on a sample space that has
equally likely outcomes.
Sample Space – All possible
outcomes.
P(event) =
Favorable outcomes
Possible outcomes
Example of Theoretical Probability…
Find the theoretical probability
of getting an even number when
you roll a number cube.
3 = 1
6
2
Examples…
You select a number at random
from the sample space {1,2,3,4,5}.
1
Answer
Find
P(2).
Is this an
5
example of
3
theoretical or
Answer
Find
P(prime).
experimental
5
probability?
4Answer
Find P(<5).
5
Examples…
In a class of 19 students, 10 study
Spanish, 7 study French, and 2 study
both Spanish and French. One student
is picked at random.
Is this an
Answer
Find P(neither).
0
example
of
theoretical or
experimental
Find
P(French).
probability?
2
Answer
Find P(both). 19
9
Answer
19
Examples…
In a telephone survey of 150 households,
75 respondents answered “Yes” to a
particular question, 50 answered “No”, and
25 were “Not Sure”.
Find P(Yes).
Is this an
example of
theoretical or
experimental
probability?
75
Answer1
=
150 2
125
5
Answer
Find P(not Not Sure).
=
150 6
Examples…
A wallet contains four bills with denominations
of $1, $5, $10, and $20. You choose two of the
four bills from the wallet at random and add
the dollar amounts.
What is the sample space.
1
1
1
1
5
5
5
5
1
5
10
20
1
5
10
20
2
6
11
21
10
15
25
10
10
10
10
20
20
20
20
1
5
10
20
1
5
10
20
20
30
40
$2
$6
$11
$21
$10
$15
$25
$20
$30
$40
A wallet contains four bills with denominations
of $1, $5, $10, and $20. You choose two of the
four bills from the wallet at random and add
the dollar amounts.
Find P($50). 0Answer
3
Find P(at least $25). Answer
10
Probability of Multiple Events
Dependent Events- when the
outcome of one event affects
the outcome of a second event.
Independent Events- when the
outcome of one event does not affect
the outcome of a second event.
Let’s Classify Events
Roll a number cube.
Then toss a
Dependent
coin.
Pick a flower from a garden.
Then pick another
flower from
Independent
the same garden.
Probability of A and B– If A and
B are independent, then
P(A and B) = P(A)P(B)
Example…
A jar contains 6 blue marbles and 8 yellow
marbles. Find the probability of selecting a
yellow marble and then tossing a coin and
getting tails.
P(yellow and tails) = P(yellow)P(tails)
1
8
P(yellow and tails) =

Independent
2
14
2
P(yellow and tails) =
7
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.You replace
this pair and then choose another
pair of socks. What is the probability
that you will choose the red pair of
socks both times? 1
25
A coin is tossed and a single 6sided die is rolled. Find the
probability of landing on the
head side of the coin and rolling
a 3 on the die. 1
12
Mutually Exclusive Events- when
two events cannot happen at the
same time. (they have no outcomes
in common).
Non-mutually Exclusive Eventswhen two events can happen at the
same time. (they have one or more
outcomes in common).
Mutually Exclusive or Not…
mutually
A single 6-sided die is Not
rolled.
What is the
probability of rolling exclusive…they
an odd number or an
can occur at
even number?
the same time.
A single 6-sided die is rolled.
What is the
Mutually
probability of rolling a 5 or an odd number?
exclusive…they
cannot occur at
the same time.
Your turn…
Is the following scenario mutually
exclusive?
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?
Answer
Yes
Your turn…
Is the following scenario mutually
exclusive?
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?
Answer
No
Your turn…
Is the following scenario mutually
exclusive?
A single letter is chosen at
random from the word
SCHOOL. What is the
probability of choosing an S
or an O?
Answer
Yes
Determining Probability of Mutually
Exclusive Events
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)
Determining Probability of Mutually
Exclusive Events
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 that it is
yellow or green?
P(Yellow or Green) = P(Yellow) + P(Green)
4
3
P(Yellow or Green) =
+
10
10
P(Yellow or Green) = 7
10
Determining Probability of Mutually
Exclusive Events
A single 6-sided die is rolled. What is
the probability of rolling a 2 or a 5?
P(2 or 5) = P(2) + P(5)
1
1
P(2 or 5) =
+
6
6
1
P(2 or 5) =
3
Determining Probability of Mutually
Exclusive Events
A spinner has 4 equal sectors colored
yellow, blue, green, and red. What is the
probability of landing on red or blue
after spinning this spinner?
1
Answer
2
Determining Probability of NonMutually Exclusive Events
When two events, A and B, are nonmutually exclusive, the probability
that A or B will occur is:
P(A or B) = P(A) + P(B) - P(A and B)
Determining Probability of NonMutually Exclusive Events
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?
P(girl or A student) = P(girl) + P(A) - P(girl and A)
9
13
P(girl or A student) =
+
30
30
P(girl or A student) = 17
30
5
30
Determining Probability of NonMutually Exclusive Events
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?
Answer
.26
Dependent Probability
When two events, A and B, are
dependent, the probability of both
occurring is:
P(A and B) = P(A) · P(B|A)
In a shipment of 20 computers, 3 are defective. Three computers
are randomly selected and tested. What is the probability that all
three are defective if the first and second ones are not replaced
after being tested?
Dependent Probability
Mr. Burger needs two students to help him
with a science demonstration for his class of
18 girls and 12 boys. He randomly chooses
one student who comes to the front of the
room. He then chooses a second student
from those still seated. What is the probability
that both students chosen are girls?
P(student 1 and student 2) = P(1) · P(2|1)
18
17
P(student 1 and student 2) =
·
30
29
P(student 1 and student 2) = 306
870
Dependent Probability
In a shipment of 20 computers, 3 are
defective. Three computers are randomly
selected and tested. What is the probability
that all three are defective if the first and
second ones are not replaced after being
tested?
Answer
6