Probability
Likelihood of an event occurring!
Random Circumstances
A random circumstance
is one in which the
outcome is
unpredictable.
Test results are positive, but
does that guarantee the
person has the disease.
Someone’s name in class will
be drawn from a hat to
answer a question.
Flipping a Coin
What are the outcomes
when flipping a fair
coin?
What is the likelihood
of each event?
If we flip the coin again,
does the result of the
first flip influence the
result of the 2nd flip?
Rolling a die
What are the possible
outcomes when rolling a
fair six-sided die?
What is the likelihood of
each of the outcomes?
What is the sum of all the
events’ probabilities?
If we rolled the die 600
times, what type of
distribution would we
see? Sketch graph.
Vocabulary
Event
Sample space
Relative frequency
Law of Large Numbers
Conditional probability vs. Independent Events
Sample Space
Rolling a die…{ 1, 2, 3, 4, 5, 6 }
Flipping two coins…{HH, HT, TH, TT}
List all possible outcomes from event
Conditions for Valid
Probabilities
Each probability of an
event has value from 0
to 1.
Sum of all probabilities
of events in a sample
space is 1.
P(tails) = .5
P(roll a 4) = 1/6
P(today is Friday) = _?_
P(heads) + P(tails) = 1
More Vocabulary
Complementary Events:
complement of event A is
AC, because
P(A) + P(AC) = 1
Complementary event
of obtaining tails when
flipping a coin is the
event of obtaining
heads.
Mutually Exclusive
Events: 2 events are
disjoint because they do
not contain any of the
same events and can not
occur simultaneously.
The set of events where
all lottery digits are the
same is mutually
exclusive of the set of
events of the first and
last lottery digits being
different.
More Vocabulary
Independent Events:
knowing one event has
occurred does not
change the probability of
the other event
occurring.
Rolling a fair dice is an
example of independent
events. Knowledge of
what the first roll was has
no influence on the next
roll.
Dependent Events:
knowing one event has
occurred changes the
probability of the other
event occurring.
Conditional Probability
If I were to randomly select
students in class of 100
students without
replacement, the
probability of you being
selected changes with each
name called. 1/100, 1/99,
1/98,…0.
Independent Events
P( roll 6 and roll 6) = P(roll 6)P(roll 6)
= (1/6)(1/6)
= 1/36
Dependent Events
P( 7and 6 w/no replacement) =
= P(7 )P(6  given 7  was drawn)
= (1/52)(1/51)
Conditional Probability
Find the probability of drawing a red ball
out of a box containing 3 red, 4 blue,
and 1 white GIVEN that a blue ball
has been drawn and not replaced.
P(R|B) = 3/7
The probability of having a certain
disease is .05. The probability of testing
+ if you have the disease is .98; the
probability of testing + when you do not
have it is .10. What is the probability that
you have the disease if you test +?
P D and  
P D |   
P  

.05.98 

.05.98  .10 .95
 .3403
Addition Rule
Find the probability that you draw either an ace or a red card.
P(Ace or Red) = P(Ace) + P(Red) – P(Red Ace)
= (4/52) + (26/52) – (2/52)
= 28/52
Mutually exclusive (Disjoint): P(Ace and King) = 0
P(Ace or King) = P(Ace) + P(King) – P(Ace and King)
= (4/52) + (4/52) - 0
= 8/52
Rules
0  P(event)  1
Complementary events P(eventC) = 1 – P(event)
P(S) = 1, where S is Sample Space
If A and B are independent events,
then P(A and B)=P(A)P(B)
Conditional: If A and B are dependent events,
then P(A and B)=P(A)P(B given A occurred)
P(A and B) = P(A)P(B|A)
Sample Exam Questions
Who is more likely, male or female, to take 2
tablets?
Tablets Female Male
1
2
3
Total
5
2
4
6
10
3
2
2
4
4
0
1
1
Total
8
12
20
What is the probability that a person took only
1 tablet?
Question
Which of the following is something we
would know for sure?
a)
b)
c)
d)
e)
How likely any outcome is to occur
When a specific outcome will occur
How often an outcome will occur
We don’t know any of the above
We would know a, b, and c.
How can we determine
probabilities?
 Through simulation.
 From random sampling in an experiment.
 From a mathematical formula/equation.