MBAD 51415142

1
MBAD 5141/5142

• Probability

2
Outline for todays class

• Basic Definitions
• Probability Models
• Conditional Probabilities

3
Probability Definitions
Probability is the scientific study of
uncertainty. While it began as a study of games
of chance (i.e. poker, craps, blackjack, etc.) it
has evolved into more diverse settings such as
medicine, engineering, and business. It is
especially useful in business forecasting and
quality assurance. To understand probability,
some definitions need to be introduced.
4
Sample Space
The SAMPLE SPACE is the set of all possible
outcomes of an experiment. For example, if you
toss a coin the experiment is tossing the coin
and the possible outcomes are heads or tails.
Each element of the sample space is called the
sample point. Sample Space is denoted by S.
5
Sample Space (continued)
Sample Spaces can be drawn out or illustrated in
several ways

• As a set. For example, if the experiment is
  tossing two coins then the sample space can be
  written as HH, HT, TH, TT.

• As a list. For example, rolling two six sided
  dice would yield the sample space 1,1 1,2 1,3
  1,4 1,5 1,6 2,1 2,2 2,3 ..6,6

6
Sample Space (continued)
Another way to write out a sample space is as a
tree diagram. Consider, again, the experiment of
tossing two coins
2nd Coin
1st Coin
HH
H
HT
start
TH
T
TT
7
Probability Definitions (continued)
An EVENT is a subset of a sample space S for a
particular experiment. EVENTS are denoted by E.
Consider the following example
Roll a six sided die with each side containing
the number 1,2,3,4,5, and 6 respectively. An
EVENT (E1) could be the times the die shows an
even number. Another EVENT (E2) could be the
times 3 or 6 show.
8
Probability Definitions (continued)
Sample Spaces and Events are often illustrated
with VENN Diagrams. In a Venn diagram the sample
space S itself is represented by a number of
points enclosed within a rectangular boundary,
each of the points representing one of the
outcomes of the experiment. Consider, again, the
experiment of tossing two coins.
Suppose we are interested in E1HT,TH (i.e.,
the event that both coins fall differently) and
in E2 TT,TH (i.e., the event that the first
coin falls tails)
HH
HT
E1
E2
TH
TT
9
Probability Definitions (continued)
Consider, again, the experiment of tossing two
coins.
Suppose we assign E1HT,TH to be the event that
both coins fall differently and E2 TT,TH to be
the event that the first coin falls tails.
Suppose we are now interested in the event that
either the first coin falls tails or else both
coins fall differently. This will occur whenever
E1 happens or E2 happens or both E1 and E2 happen
together. It will be called E3HT,TH,TT. Notice
that this event has some elements common to both
E1 and E2.
HH
E1
HT
E2
TH
TT
10
Unions and Intersections
Let E1 and E2 be two events of a sample space S.
then the UNION of two events E1 and E2, denoted by
Is the set of all the sample points (elements) in
E1 or E2 or both.
E2
E1
11
Unions and Intersections
Let E1 and E2 be two events of a sample space S.
then the INTERSECTION of two events E1 and E2,
denoted by
Is the set of all the sample points (elements)
that belong to both E1 and E2 at the same time.
E2
E1
12
Mutually Exclusive
Two events E1 and E2 are said to be mutually
exclusive if there is no sample point which is in
both E1 and E2. Stated symbolically
E1
E2
13
Complement
Let E be an event in a sample space D. Then the
complement E of the event E with respect to the
sample space S is the set of all outcomes in S
which are not in E.
E
E
14
Probability
The Probability of a particular outcome indicates
how likely that outcome will occur. Probability
is based on the theory that, over the long term,
a proportion of successes will occur. Probability
is based on studying randomness and the patterns
that occur by chance when an act is committed
over and over. Randomness, as it applies to
probability, is not a series of haphazard events.
Rather , it is a description of a kind of order
that emerges only in the long run. This is an
important concept in the study of statistical
analysis.
15
Probability
Probability can be ascertained by forming a ratio
of successful events to the sample space. For
example, if an experiment is conducted in which
two coins are tossed and E1HH, then the
probability of E1 occurring is ¼. This is because
there is only one possible occurrence of E1 out
of a sample space of 4 elements HH, HT, TH, TT.
If we assign another event E2 HT, TH then the
probability of E2 occurring is ½. Can you tell
why?
16
Probability
Probability is denoted by P. For example, the
probability of event A occurring is P(A).
Probability also follows these basic rules
Rule 1 Any probability is a number between 0
and 1. Rule 2 All possible outcomes together
must have a probability of 1.Rule 3 The
probability that an event does not occur is 1
minus the probability that the event does occur.
This is known as the complement rule and is
stated as P(A')1- P(A).Rule 4 If two events
have no outcomes in common (i.e., they are
disjoint), the probability that one or the other
occurs is the sum of their individual
probabilities. In notation form this is P(A or
B) P(A) P(B) P(A B)and is known as the
addition rule for disjoint events.
17
Probability Example 1

• A card is drawn from a well-shuffled pack of 52
  cards. Find the following probabilities
• A Heart
• A 2 or a 9
• Not a Spade
• A 6 or a Diamond

18
Probability Example 2

• Let E1 be the event that a stock will rise next
  year and E2 the event that its price will remain
  unchanged. If P(E1) 0.3 and P(E2) 0.45,
  calculate the following
• P(E2)
• P(E1 or E2)
• P(E1 and E2)

19
Conditional Probability
Additional information about the outcome of an
experiment can change the probability of
associated events. Consider a deck of cards. If
an experiment is to draw one card and E1drawing
a 10 of spades, then P(E1)1/52. If the card is
placed back into the deck and E2drawing a 5 of
diamonds, then P(E2)1/52 also. But what if the
experiment is changed to drawing two cards
without replacement and suppose that the first
card drawn is a King of Clubs. Then P(E2)1/51.
In other words, the probability of drawing a 5 of
diamonds out of a deck is conditional on the
result of the first draw. This brings up another
definition.
20
Conditional Probability
Let E1 and E2 be two events. Then the conditional
probability of E2 given E1, written as P(E2E1),
is the probability that E2 occurs given that E1
is known to have occurred.
P(E2E1)
P(E1E2)
and
These can also be written as
P(E1
E2) P(E1)P(E2E1)
21
Independence
Two events E1 and E2 are independent if the
probability of either one occurring does not
depend on whether or not the other occurs. That
is, E1 and E2 are independent if
22
One More Probability Example
An automobile manufacturer buys computer chips
from a supplier. The supplier sends a shipment
containing 5 defective chips. Each chip chosen
from the shipment has a probability 0.05 of being
defective, and each automobile uses 12 chips
selected independently. What is the probability
that all 12 chips in a car will work properly?
23
Okay, Another Probability Example
Functional Robotics Corporation buys electrical
controllers from a Japanese supplier. The
companys treasurer thinks that there is a
probability of 0.4 that the dollar will fall
against the Japanese yen in the next month. The
treasurer also believes that if the dollar falls
there is a probability of 0.8 that the supplier
will demand renegotiation of the contract. What
is the probability that the dollar falls and the
supplier demands renegotiation of the contract?