Transcript Sample Space, S

Sample Space, S
• The set of all possible outcomes of an experiment.
• Each outcome is an element or member or
sample point.
• If the set is finite (e.g., H/T on coin toss, number
on the die, etc.):
– S = {H, T}
– S = {1, 2, 3, 4, 5, 6}
– in general, S = {e1, e2, e3, …, en}
• where ei = the outcomes of interest
• Note: sometimes a tree diagram is helpful in
determining the sample space…
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Sample Space
• Example: The sample space of gender and
specialization of all BSE students in the School
of Engineering is …
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Events
• A subset of the sample space reflecting the
specific occurrences of interest.
• Example,
– All female students,
F=
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Events
• Complement of an event, (A’, if A is the event)
– e.g., students who are not female,
• Intersection of two events, (A ∩ B)
– e.g., engineering students who are EVE and female,
• Mutually exclusive or disjoint events
• Union of two events, (A U B)
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Venn Diagrams
• Example, events V (EVE students) and F
(female students)
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Other Venn Diagram Examples
• Mutually exclusive events
• Subsets
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Example:
• Students who are male, students who are ECE,
students who are on the ME track in ECE, and
female students who are required to take ISE
412 to graduate.
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Sample Points
• Multiplication Rule
– If event A can occur n1 ways and event B can occur
n2 ways, then an event C that includes both A and B
can occur
n1 n2
ways.
– Example, if there are 6 ways to choose a female
engineering student at random and there are 6 ways
to choose a male student at random, then there are
6 * 6 = 36
ways to choose a female and a male engineering
student at random.
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Another Example
• Example 2.14, pg. 32
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Permutations
• definition: an arrangement of all or part of a set
of objects.
• The total number of permutations of the 6
engineering specializations in MUSE is …
• In general, the number of permutations of n
objects is
n!
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Permutations
• If we take the number of specializations 3 at a
time (n = 6, r = 3), the number of permutations is
• In general,
n!
n Pr 
n  r !
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Example
• A new group, the MUSE Ambassadors, is being formed
and will consist of two students (1 male and 1 female)
from each of the BSE specializations. If a prospective
student comes to campus, he or she will be assigned
one Ambassador at random as a guide. If three
prospective students are coming to campus on one day,
how many possible selections of Ambassador are there?
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Combinations
• Selections of subsets without regard to order.
• Example: How many ways can we select 3
guides from the 12 Ambassadors?
n
n!
  

 r  r ! n  r !
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Probability
• The probability of an event, A is the likelihood of
that event given the entire sample space of
possible events.
0 ≤ P(A) ≤ 1
P(ø) = 0
P(S) = 1
• For mutually exclusive events,
P(A1 U A2 U … U Ak) = P(A1) + P(A2) + … P(Ak)
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Calculating Probabilities
•
Examples:
1. There are 26 students enrolled in this section of EGR
252, 3 of whom are BME students. The probability of
selecting a BME student at random off of the class roll
is:
P = ______________________
2. The probability of being dealt 2 aces & 3 jacks in a 5card poker hand is:
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Additive Rule Examples
1. Draw 1 Card. Note Kind, Color & Suit.
– Probabilities associated with drawing an ace and
with drawing a black card are shown in the following
contingency table:
Color
Type
Ace
Red
Black
Total
2
2
4
Non-Ace
24
24
48
Total
26
26
52
– Therefore the probability of drawing an ace or a
black card is given by:
4 26 2
28
P ( A  B )  P ( A )  P (B )  P ( A  B ) 



52 52 52 52
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Additive Rules
2. After the first card is drawn, it is returned to the deck
which is shuffled. Another card is drawn. What is the
probability that at least one of the cards is an ace?
P ( A1  A2 )  P ( A1 )  P ( A2 ) 
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4
8


52 52 52
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Applications of Probability
• Example:
An appliance manufacturer has learned of an increased incidence
of short circuits and fires in a line of ranges sold over a 5 month
period. A review of the FMEA data indicates the probabilities that if
a short circuit occurs, it will be at any one of several locations is as
follows:
Location
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House Junction
0.46
Oven/MW junction
0.14
Thermostat
0.09
Oven coil
0.24
Electronic controls
0.07
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Applications of Probability
• The probability that the short circuit does not occur at the
house junction is …
• The probability that the short circuit occurs at either the
Oven/MW junction or the oven coil is …
• The probability that both the electronic controls and the
thermostat short circuit simultaneously is …
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Your turn …
• Problem 3, page 46
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Serial and Parallel Systems
• For increased safety and reliability, systems are
often designed with redundancies. A typical
system might look like the following:
0.88
C
0.95
0.9
0.97
A
B
E
0.85
D
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Serial and Parallel Systems
• What is the probability that:
– Segment 1 works?
0.88
C
0.95
0.9
A
B
0.97
E
0.85
D
1
2
– Segment 2 works?
– The entire system works?
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Conditional Probability
0.88
C
0.95
0.9
0.97
A
B
E
0.85
1
D
2
• If segment 2 works, what is the probability that
component C does not work?
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Conditional Probability
0.88
C
0.95
0.9
0.97
A
B
E
0.85
D
• If the system works, what is the probability that
component D does not work?
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Your turn …
• Problem 3, page 54
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