Download Experiments, Outcomes, Samples Spaces, and Events

Announcements
Finite Probability
Friday, September 30th
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MyMathLab 4 is due Wednesday Oct 5
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Problem Set 4 is due Friday Oct 7
Today: Sec. 6.1: Experiments, Outcomes, Sample Spaces, Events
Describe the sample space and events of an experiment
Use set operations to define new events
Next Class: Sec. 6.2: Assignment of Probabilities
Cherveny
Sept 30
Math 1004: Probability
Definitions
Definition
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An experiment is an activity with an observable result.
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An outcome is a possible result of an experiment.
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The sample space of the experiment is the set of all possible
outcomes.
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An event is a subset of the sample space.
Cherveny
Sept 30
Math 1004: Probability
Committee Example
Example
A committee of two people is to be selected from five people,
Renee, Stacy, Travis, Ursula, and Victor (R,S,T,U,V).
(a) What is the sample space for this experiment?
Answer: {RS, RT , RU, RV , ST , SU, SV , TU, TV , UV }
(b) Describe the event “Renee is on the committee” as a subset of
the sample space.
Answer: {RS, RT , RU, RV }
(c) Describe the event “neither Renee nor Stacy is on the
committee” as a subset of the sample space.
Answer: {TU, TV , UV }
Cherveny
Sept 30
Math 1004: Probability
Events and Sets
Since events are subsets of the sample space (our “universe” of
outcomes), we may use set operations to define new events.
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Union: E1 ∪ E2 occurs when E1 or E2 (or both) occur
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Intersection: E1 ∩ E2 occurs when both E1 and E2 occur
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Complement: E 0 occurs when E does not
Cherveny
Sept 30
Math 1004: Probability
Coin Toss
Example
An experiment consists of tossing a coin three times and observing
the sequence of heads and tails.
(a) What is the sample space of this experiment?
(b) Determine the event E1 = “more heads than tails occur”
(c) Determine the event E2 = “the first toss is a head”
(d) Determine the event E1 ∩ E2
(e) Determine the event E10
(f) Determine the event (E1 ∪ E2 )0
Cherveny
Sept 30
Math 1004: Probability
Practice
1. A coin is tossed and a die is rolled, and both results are
observed. How many outcomes are in the sample space?
2. A letter is selected at random from the word “MISSISSIPPI”
(a) What is the sample space for this experiment?
(b) Describe the event “the letter chosen is a vowel” as a subset of
the sample space.
3. An orange and a black six-sided die are rolled, and the results
facing up are observed.
(a) List the sample space, or just write out a few examples and say
how many elements there are.
(b) Describe each of the following events:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Cherveny
Both numbers are even.
At least one number is even.
Neither number is less than or equal to 3.
The sum of the numbers is 6.
The sum of the numbers is greater than or equal to 8.
The numbers are the same.
A 2 or 3 occurs, but not both 2 and 3.
No 4 appears.
Sept 30
Math 1004: Probability
Practice Answers
1. 2 · 6 = 12 outcomes
2. (a) {M, I , S, P}
(b) {I }
3. (a) {(1, 1), (1, 2) . . . , (1, 6), (2, 1), (2, 2), . . . , . . . , (6, 6)}. There are
6 · 6 = 36 outcomes in the sample space.
(b)
Cherveny
(i) There are 9 outcomes in this event. When a problem asks you
to describe or list outcomes, you should give a set not just a
size... i.e. {(2, 2), (2, 4), (2, 6), (4, 2), . . . }
(ii) There are 27 outcomes in this event.
(iii) There are 9 outcomes in this event.
(iv) There are 5 outcomes in this event.
(v) There are 15 outcomes in this event.
(vi) There are 6 outcomes in this event.
(vii) There are 18 outcomes in this event.
(viii) There are 25 outcomes in this event.
Sept 30
Math 1004: Probability