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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. 41

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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

• Let’s say we want to know how many smaller arrangements of the set we can make, e.g.– If we take the number of specializations 3 at a time

(n = 6, r = 3), the number of permutations is

• In general,

!!

rn

nPrn

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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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Looking at a more complicated example …

• example 2.18, pg. 44

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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?

!!

!

rnr

n

r

n

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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 5-card poker hand is:

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Additive Rule Examples1. 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:

– Therefore the probability of drawing an ace or a black card is given by:

Type

Color

TotalRed Black

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52

28

52

2

52

26

52

4)()()()( BAPBPAPBAP

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Additive Rules2. 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?

52

8

52

4

52

4)()()( 2121 APAPAAP

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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 P

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