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An event is the set of possible outcomes
Probability is between 0 and 1
The event A has a complement, the event not A.
Together these two probabilities sum 1.
ex. At least one and none are complements
Probability of an event =
number of outcomes in event / number of equally likely outcomes
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Probability Distributions give all values
resulting from a random process.
Sample space is the complete list of disjoint
outcomes. All outcomes in a sample space
must have total probability equal to 1.
Ex. The sample space for rolling a die is
{1,2,3,4,5,6} The sample space for rolling two
die is the table of 36 outcomes we’ve seen.
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Disjoint and mutually exclusive mean the
same thing.
Disjoint means two different outcomes can’t
occur on the same opportunity
Ex. Can’t roll an extra credit and no collect on
the same roll. Can’t get a heads and tail on
the same flip.
These items on a Venn diagram would have
no intersection
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Flipping a coin and getting a head and tail is
disjoint
Flipping a coin twice and getting a head and
tail is disjoint.
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In random sampling, the larger the sample,
the closer the proportion of successes in the
sample tends to be to the population
proportion.
The difference between a sample proportion
and the population proportion must get
smaller as the sample size gets larger.
Processes can be split into stages (Flip coin
once, then again)
 If there are k stages with different possible
outcomes for each stage, the number of total
possible outcomes is
n1*n2*n3*n4… nk
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Ex. How many total outcome for rolling a die
and flipping a coin?
6*2 = 12
1)
Assumptions
What probability are you assuming? Are
events independent?
2) Model
*How specifically will you use your table of
random digits?
*Make sure to say what to do with repeats,
unallocated numbers
*Fully describe what constitutes a run and
what statistics you’re collecting.
*Make a table to show how digits /groups
are assigned
3) Repetition
Run the simulations and record the results
in a frequency table.
4) Conclusion
Write the conclusion in context of the
situation. Be sure to say the probability is
ESTIMATED.
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Remember first that “or” means one or the
other or both
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P(A or B) = P(A) + P(B) – P(A and B)
If A and B are disjoint, there is no
intersection. Therefore, P(A and B)= 0.
If A and B are disjoint: P(A or B) = P(A) + P(B)
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The probability of an event A and B both
happening is
P(A and B) = P(A) * P(B|A)
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P(A and B) = P(B) * P(A|B)
The probability of an event changes based on
what happened before.
Rearranging means
P(A|B) = P(A and B) / P(B)
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Probability of both divided by the probability
of the first event.
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The occurrence of one event doesn’t change
the probability of the second event occurring
Test for Independence:
IS P(A|B) = P(A) or P(B|A) = P(B) ?
If yes, you have independent events.
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Sometimes this isn’t obvious that one has an
effect without checking
Remember if events are independent,
P(A|B) = P(A) or P(B|A) = P(B)
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Therefore, since P(A and B) = P(A) * P(B|A)
P(A and B) = P(A) * P(B)
for independent events
If events A and B are independent and P(A) =
0.3 and P(B) = 0.5, then which of these is
true?
A. P(A and B) = 0.8
B. P(A or B) = 0.15
C. P(A or B) = 0.8
D. P(A | B) = 0.3
E. P(A | B) = 0.5
If events A and B are independent and P(A) = 0.3
and P(B) = 0.5, then which of these is true?
A. P(A and B) = 0.8
B. P(A or B) = 0.15
C. P(A or B) = 0.8
D.
P(A | B) = 0.3
Probability of A is not changed
based on the occurrence of event B
E.
P(A | B) = 0.5
Crash Type
Single Vehicle
Multiple
Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol
Related
11,345
11,336
22,681
Total
22,086
16,223
38,309
If a fatal auto crash is chosen at random, what is the approximate probability
that the crash was alcohol related, given that it involved a single vehicle?
A.
0.28
B.
0.49
C.
0.58
D.
0.69
E.
The answer cannot be determined from the information given.
Crash Type
Single Vehicle
Multiple
Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol
Related
11,345
11,336
22,681
Total
22,086
16,223
38,309
If a fatal auto crash is chosen at random, what is the approximate probability
that the crash was alcohol related, given that it involved a single vehicle?
B. 0.49
10,741/22,086 = 0.49
Crash Type
Single Vehicle
Multiple
Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol
Related
11,345
11,336
22,681
Total
22,086
16,223
38,309
What is the approximate probability that a randomly chosen fatal auto crash
involves a single vehicle and is alcohol related?
A.
0.28
B.
0.49
C.
0.58
D.
0.69
E.
The answer cannot be determined from the information given
Crash Type
Single Vehicle
Multiple
Vehicles
Total
Alcohol Related
10,741
4,887
15,628
Not Alcohol
Related
11,345
11,336
22,681
Total
22,086
16,223
38,309
What is the approximate probability that a randomly chosen fatal auto crash
involves a single vehicle and is alcohol related?
A. 0.28
Because 10,741/38,309 = 0.28
For all events A and B, P(A and B) =
A. P(A) · P(B)
B. P(B | A)
C. P(A | B)
D. P(A) + P(B)
E. P(B) · P(A | B)
For all events A and B, P(A and B) =
A. P(A) · P(B)
B. P(B | A)
C. P(A | B)
D. P(A) + P(B)
P(B) · P(A | B)
This is the multiplication rule.
If they are independent,
P(A|B) = P(A)
E.
The mathematics department at a school has
twenty instructors. Six are easy graders. Twelve
are considered to be good teachers. Seven are
neither. If a student is assigned randomly to one
of the easy graders, what is the probability that
the instructor will also be good?
A. 7/20
B. 5/12
C. 7/12
D. 5/6
E. The answer cannot be determined from the
information given.
D.
5/6
Easy
Not Easy
Total
Good
5
7
12
Not Good
1
7
8
Total
6
14
20
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The management of Young & Sons Sporting Supply, Inc., is
responding to a claim of discrimination. If the company has
employed the 60 people in this table, how many females
over the age of 40 must the company hire so that the age
and sex of its employees are independent?
40 Years
> 40 Years
Male
25
15
Female
20
Total
Total
First, let N be the number of females older than
40 to be hired.
40 Years > 40 Years
Total
Male
25
15
40
Female
20
N
20+N
Total
45
15+N
60+N
Set up a proportion so
P(male |<40) = P(Female|< 40)
N=12
If P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.08, which of
these is true?
A.
Events A and B are independent and mutually
exclusive.
B.
Events A and B are independent but not mutually
exclusive.
C.
Events A and B are mutually exclusive but not
independent.
D.
Events A and B are neither independent nor
mutually exclusive.
E.
Events A and B are independent, but whether A
and B are mutually exclusive cannot be
determined from the given information.
If P(A) = 0.4, P(B) = 0.2, and P(A and B) = 0.08, which of
these is true?
A.
Events A and B are independent and mutually
exclusive.
B.
C.
D.
E.
Events A and B are independent but not
mutually exclusive.
Events A and B are mutually exclusive but not
independent.
Events A and B are neither independent nor
mutually exclusive.
Events A and B are independent, but whether A
and B are mutually exclusive cannot be
determined from the given information.