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Sample Spaces and Events
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Experiments
Sample spaces and Events
Probability of Event
Equally likely assumption
Random experiments:
Experiments which do not yield the same results ( flipping coins, rolling dice, spinning a
number wheel, ..) The single outcome from an experiment cannot be predicted with
certainty.
Sample spaces and events:
The sample space (S) is the collection of basic outcomes of an experiment.
Example : Rolling a die
Sample space: { 1, 2, 3, 4, 5, 6}
An event E is defined to be any subset of S (including the empty set and the sample space
S). Event E is a simple event if it contains only one element and a compound event if it
contains more than one element.
Example:
The outcome is any number less than 2: {1} - simple event
The out come is any prime number: {1, 3, 5} – compound event
(Define a prime number)
Example:
A number wheel of 18 numbers from 1-18 is spun, and allowed to come to rest so that a
pointer points within a numbered sector.
Experiment : Observe the number at which the pin points when the wheel stops.
Sample space : { 1, 2, 3, 4, …., 18}
What is the event E that corresponds to each of the following outcomes? Indicate whether
the event is a simple event or a compound event.
(a) The outcome is a number divisible by 12.
(b) The out come is an even number greater than 15.
With the same experiment, the sample space can be different, depending on what in the
interest is.
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How to define a sample space?
Example:
1. A nickel and a dime are tossed. The way to identify a sample space for this experiment
depends on our interest.
(a) If we are interested in whether each coin falls heads (H) or tails(T), then
S1=
(b) If we are interested only number of heads that appear on a single toss of the two
coins,
S2 =
(c) If we are interested in the coins match (M) or not match (D), then
S3 =
Choose the sample space that includes as much detail as necessary to answer all
questions of interest, regarding to the outcomes of the experiment.
2. An experiment consists of recording the boy-girl composition of 2-child family.
(a) What is an appropriate sample space if we are interested in the sex of each child
in the order of their births? Draw a tree diagram.
(b) What is an appropriate sample space if we are interested only in the number of
girls in a family?
(c) What is an appropriate sample space if we are interested only in whether the sexes
are alike (A) or different (D).
(d) What is an appropriate sample space for all three interests expressed in parts a-c?
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3. Rolling two dice.
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2
3
4
5
6
1
2
3
4
5
6
What is the event that corresponds to following outcomes?
(a) A sum of 7 turns up
(b) A sum of 11 turns up
(c) A sum of less than 4 turns up
(d) A sum of 5 turns up.
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3. Probability of an event
Probability for simple events
Given S = {e1, e2,…, en}, with n simple events.
P(ei) – probability of the event ei, under the following conditions:
**1. 0  P(ei )  1
**2. P(e1 )  P(e2 )    P(en ) = 1
Any probability assignment that meets conditions 1 and 2 is said to be an acceptable
probability assignment.
Reasonable assignments ---based on expected or actual percentage of times an event
occurs when the experiment is repeated a large number of times.
Tossing a coin: S = {
}; P(H) =
P(T) =
Probability of an event (E) – given an acceptable probability assignment
(A) If E is empty, P(E) =
(B) If E is a simple event, P(E) has already been assigned.
(C) If E is a compound event, then P(E) is the sum of the probabilities of all
the simple event in E.
(D) If E is the sample space, P(E) = P(S) =
.
Example
1. Rolling a fair die
Simple
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event (ei)
P(ei)
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3
4
5
6
(a) What is the probability of getting 1?
(b) What is the probability of getting a number less than 4?
(c) What is the probability of getting a number greater than 6?
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2. Suppose after rolling a die 1,000 times, we find that 1 turns up 100 times, 2 turns up
100 time, 3 turns up 200 time, 4 turns up 200 time, 5 turns up 150 and 6 turns up 250
times.
(a) Assign probabilities to simple events in S
Simple
event (ei)
P(ei)
1
2
3
4
5
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(b) Find probabilities of following events.
a. E1 = getting 1
b. E2 = Getting a number less than 4
c. E3 = Getting either 1 or 6
There are two ways of construction acceptable and reasonable probability assignment.
1. Theoretical
2. Empirical
Probability of a simple event under an equally likely assumption
If, in a sample space S= { e1, e2, … , en } with n elements, we assume each simple
event ei is as likely to occur as any other, then we assign the probability 1/n to each.
That is
P(ei) = ____________.
Example
If a single die is rolled and we assume each face is as likely to come up as any other,
S = _______________________________________
E1 = getting number 1 = {
E2 = getting an even number = {
}
}
P(E) = ____________________________
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Conclusion: If we assume each simple event in a sample space S is as likely to occur
as any other, then the probability of an arbitrary event E in S is given by
P(E) =
Example
1. Consider rolling two dice, and assume each simple event in the sample space is as
likely as any other.
1
2
3
4
5
6
1
2
3
4
5
6
Find the probabilities of the following event.
(a) A sum of 7 turns up
(b) A sum of 11 turns up
(c) A sum of less than 4 turns up
(d) A sum of 5 turns up.
2. In drawing 7 cards from a 52-card deck without replacement, what is the
probability of getting 7 hearts?
3.The board of regents of a university is made up of 12 men and 16 women. If a
committee of 6 chosen at random, what is the probability that it will contains 4 men
and 2 women.
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