Download Sample Spaces, Events, and Counting Techniques Random

Chapter 2: PROBABILITY
Part 1: Sample Spaces, Events,
and Counting Techniques
Section 2.1
Random Experiments
• A random experiment is an experiment that
can result in a different outcome, even though
the experiment is repeated under the same
conditions.
– Bake cake with a given recipe, at 350◦, in
a certain oven, for a certain time length,
measure moisture, repeat, ...
• Goal is to understand, quantify, and model
the type of variations that we encounter.
– What might be causing the differences in
the levels of moisture for the cakes?
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• In experiments, some variables are controlled,
and others are not.
– Sometimes, we realize (after the fact), that
we should have controlled some of the variables we left uncontrolled.
– But we can’t control everything, so we
have to make good choices.
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• Example process:
Catapult a large ball bearing
Response: Distance ball bearing travels
Controlled variables?
– weight of ball bearing
– angle of release
– force at release
Uncontrolled variables?
– air resistance/wind/environmental conditionals
– small differences in bearings
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Sample Spaces
• A sample space denoted as S is a set or collection of outcomes of a particular random
experiment.
(Describes all the things that could happen).
If we want to discuss which outcomes are
more or less likely to occur, we first need to
specify all the possible things that could occur, i.e. we should define the sample space.
1. Consider tossing a coin once. S = {H, T }
2. Consider rolling a die. S = {1, 2, 3, 4, 5, 6}
3. Consider an experiment in which you count
the number of defects on a molded part.
S = {0, 1, 2, 3, . . .}
4. Consider an experiment where you measure the thickness of a part.
S = R+ = {x|x > 0}
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5. Consider an experiment where you measure the thickness of two manufactured
parts. S = R+ × R+
6. A batch contains 4 items: a, b, c, d.
Consider the unordered sample space
for selecting two items without replacement. S = {ab, ac, ad, bc, bd, cd}
(unordered means order doesn’t matter,
i.e. we consider ab the same as ba)
• A sample space is discrete if it consists of
a finite or countable infinite set of outcomes
(like the numbers 0, 1, 2, 3,...).
• A sample space is continuous if it contains
an interval (either finite or infinite) of real
numbers.
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Distinguishing between discrete and continuous sample spaces...
Continuous:
– The thickness of a part.
S = R+ = {x|x > 0}
– The weight of a child.
S = R+ = {x|x > 0}
Discrete:
– Tossing a coin once. S = {H, T }
– An experiment in which you count the number of defects on a molded part.
S = {0, 1, 2, 3, . . .}
Note that even though there’s no limit
to the number of defects above, it is still
discrete.
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Discrete (continued):
– The weight of a child measured by a scale
that rounds to the nearest tenth of a pound.
S = {0.0, 0.1, 0.2, ..., X} where X is the
scale maximum.
Even though ‘weight’ is theoretically continuous, the weight provided by the scale
is discrete because of the rounding process.
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• Describing sample spaces graphically with
tree diagrams.
1. There are 4 options on vehicles from a manufacturer:
(a)
(b)
(c)
(d)
With or without automatic transmission
With or without air conditioning
One of three choices of a stereo system
One of four exterior colors
Let S represent the set of all possible vehicle types.
Every node at the bottom is a different combination.
There are 2 × 2 × 3 × 4 = 48 combinations.
S is represented by the full set of 48 nodes at the bottom.
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Events
• An event is a subset of the sample space S of
a random experiment. It is a set of possible
outcomes taken from S.
1. Consider the batch of 4 items: a, b, c, d.
Suppose we choose 2 items without replacement and order doesn’t matter,
S = {ab, ac, ad, bc, bd, cd}.
Let E1 be the subset of outcomes for which
a is chosen. E1 = {ab, ac, ad}. E1 is an
event.
2. Consider the sample space for rolling a die,
S = {1, 2, 3, 4, 5, 6}.
• Let E1 = {2, 4, 6}, the subset coinciding with the event of rolling an even
number.
• Let E2 = {1, 3, 5}, the subset coinciding with the event of rolling an odd number.
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Looking forward in the course...
• We often want to discuss the probability of
a certain occurrence (or a set of occurrences
with something in common), like...
The probability that a randomly chosen
object has a defect.
The probability that a randomly chosen
cell phone is not an iPhone.
Thus, we formally define a subset of the possible outcomes (so, we define an event) and
then mathematically calculate the probability that the event (or subset) actually occurs.
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Set Operations
• The union of the sets A and B, denoted as
A ∪ B, is the set of outcomes in either A or
B, or both A and B.
Consider E1 (evens) and E2 (odds) that were
defined earlier.
E1 ∪ E2 = S
because putting E1 and E2 together
make-up the whole sample space.
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Continuing the roll of a die, let Ei = {1, 2} and
Ej = {2, 6}. Then, Ei ∪ Ej = {1, 2, 6}.
More Set Operations
• The intersection of the sets A and B, denoted as A ∩ B, is the set of outcomes that
are common in A and B.
Using our previously defined events
E1 (evens) and E2 (odds)...
E1 ∩ E2 = ∅
These events have nothing in common.
∅ represents the empty set or null set used
when there are no elements in a defined set.
And Ei ∩ Ej = {2}
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• The complement of A, denoted as Ac or A0,
is the set of outcomes in S that are not in A.
For the odds and evens, E10 = E2
E2 is the complement of E1.
E1 ∩ E2 = ∅
E1 ∪ E2 = S
and
Using Ei = {1, 2} and Ej = {2, 6},
Ei0 = {3, 4, 5, 6}
and Ej0 = {1, 3, 4, 5}
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• Venn Diagrams
– Used to visually describe relationships between events.
In the following Venn diagrams, the shaded
region represents the set stated at the upper right of the last 3 boxes (or sample
spaces).
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• Example:
1. An engineering firm is hired to determine
if certain waterways in Iowa city are safe
for fishing. Samples are taken from three
rivers.
Let G represent an outcome where a river
was found to be safe for fishing, and F
represent an outcome where a river failed
to be safe for fishing.
i) List the elements of the (ordered) sample space S.
ii) Let E1 be the event that at least two
of the rivers are safe for fishing. List the
elements of E1.
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iii) Define (in words) an event E2 that has
the elements...
{GGF, GF G, F GG, GF F, F F G, F GF, F F F }.
iv) Define (in words and symbol) the
complement of E2.
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• Example: Exercise 2-6 p.26
2. An ammeter that displays three digits is
used to measure current in milliamperes.
Give the sample space.
• Example: Exercise 2-13 p.26
3. The time of a chemical reaction is recorded
to the nearest millisecond.
Give the sample space.
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Counting Techniques
- Sometimes, instead of writing out all the outcomes for a sample space, we instead consider the counts of the number of outcomes
for analysis.
• Multiplication Rule
(for counting techniques)
– If an operation can be described in a
sequence of k steps, and the number of
ways to complete step 1 is n1, and the
number of ways to complete step 2 is n2,
and so forth, then the total number of
ways of complete the operation is...
n1 × n2 × · · · × nk
– Example: As in the automobile maker
example, there were
2 × 2 × 3 × 4 = 48 possible vehicles
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– Example: In the game Guitar Hero,
you get to choose a character/guitar/venue
combination. You have 8 characters to
choose from, 6 guitars, and 4 venues to
choose from. There are 8 × 6 × 4 = 192
possible options.
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• Permutations
– A permutation of elements is an ordered
sequence of the elements (order matters).
– Example: Consider the set of three numbers {1, 2, 3}. Here are the possible permutations of the elements in the set:
(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)
– The number of permutations of n different
elements is n! (pronounced “n factorial”)
where
n! = n × (n − 1) × · · · × 2 × 1
In the above example, there are 3! = 3 ×
2 × 1 = 6 permutations of the set of 3
unique elements.
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3! = 3 × 2 × 1 = 6
Another way to think of it...
There are 3 ways to ‘fill’ the first slot.
After you choose the first, there are 2 ways
to ‘fill’ the second slot. And then there is
only 1 way left to fill the last slot.
– Example: Five people stand in line at
a movie theater. Into how many different
orders can they be arranged?
Ans:
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• Permutations of subsets
– Sometimes we are interested in counting
the number of permutations of subsets of
a certain size chosen from a larger set.
– Example: Five lifeguards are available
for duty. There are three lifeguard stations. In how many ways can three lifeguards be chosen and ordered among the
stations?
Five ways to choose the first, 4 ways to
choose the second, 3 ways to choose the
last station.
– In general, the number permutations of r
objects chosen from a group of n objects is
Prn = n × (n − 1) × · · · × (n − r + 1)
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– Or this can be stated in factorial notation
as
Prn = n(n − 1) · · · (n − r + 1)
n(n−1)···(n−r+1)(n−r)(n−r−1)···(3)(2)(1)
(n−r)(n−r−1)...(3)(2)(1)
n!
= (n−r)!
=
Example: Find the number of ways of selecting a president, vice president, secretary, and
treasurer in a club of 15 people.
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• Permutations of Similar Objects
– Sometimes we are interested in counting
the number of ordered sequences for objects when not all the objects are considered different (i.e. are indistinguishable).
– Example: A part is labeled with a bar
code such that 2 thin lines (t) and 3 fat
lines (f) will be used (for example, |||||).
How many distinct patterns can be created if the thick and thin lines can be put
in any order?
ANS: There are 10 distinct labelings...
ttfff, tftff, tfftf, tffft, fttff,
ftftf, ftfft, ffttf, fftft, ffftt
It might be tempting to answer 5! = 120,
but this would over count the number of
unique scanning bar codes.
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– In general, the number of permutations of
n = n1 + n2 + · · · nr objects of which n1
are of one type, n2 are of a second type,
. . ., and nr are of an rth type is...
n!
n1!n2!···nr !
(This is also called the multinomial coefficient.)
5! = 10
For the bar code example we have 2!3!
• Combinations
– In many problems we are interested in the
number of ways of selecting r objects from
n without regard to order (i.e. order doesn’t
matter). These selections are called
combinations.
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– The number of combinations, subsets of
size r that can be selected
from
a set of n
n
elements, is denoted as
or Crn and
r
pronounced “n choose r” and
n
n!
= r!(n−r)!
r
– Example: At a company with 35 engineers, the boss will be choosing 5 to go to
a conference. How many different groups
of 5 members are there to choose from?
ANS:
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If you’re looking for the number of arrangements, you’re probably considering a permutation... order matters.
Cabinet members where each position is unique,
like president, treasurer, secretary, etc.
If you’re choosing a subset where the order doesn’t
matter (like in a team of equal players or a
committee), then you’re probably considering a
combination.
Choosing committee members where all
members are of equal standing and
essentially exchangeable.
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