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IME350
Engineering Data Analysis
Lecture 3
Probability Course Packet
Slides 1-20
1
Some morbid probabilities…
• Calculating death rates for various activities is
a very valuable pursuit for those engaged in
selling life insurance…Check out a few of these
probabilities*:
Deaths per 100,000
30
25
20
Driving
15
Trains
10
Flying
5
0
1930
1940
1950
1960
1970
1980
1990
YEAR
*Numbers were obtained from various unverified sources on the web
ACR 2
• Let’s use the most current death rate of
about
– 18/100,000 for driving
– 1/100,000 for riding the train
– 1/100,000 for flying
• Now compare to some activities that we
Bozemanites are quite likely to engage in
“hiking”
death rate =
2.62/100,000
Downhill skiing
death rate =
0.52/100,000
Rafting death rate =
0.55/100,000 “user days”
Kayaking death rate =
2.9/100,000 “user days”
Climbing (rock and
ice) death rate =
6.6/100,000
ACR 3
Annapurna descent death rate = 7.3/100
K2 descent death rate = 11.6/100
All other 8000 m peak descent
death rate = 2.3/100
Keep in mind….
Odds of dying in Russian Roulette are 16.7/100!!!
ACR 4
• Now we are digging into some of the basic concepts of
probability: Sample spaces, events, probability axioms
and properties, counting techniques, conditional
probability and independence.
Probability
Sample
Population
Inferential
Statistics
5
Characteristics of probability
problems…
• Probability = the numerical measure of the
likelihood of occurrence of an event relative to
a set of alternative events.
• Implicitly there is more than one possible
outcome (or the problem would be
deterministic)
• Two things have to be defined
– Sample space
– Event of interest
6
Sample Space (S)
• The set of all possible outcomes in an
experiment
• Simplest example is a space with two possible
outcomes: Coin toss, pass-fail inspection, sex
of baby at birth, …
Pass
Fail
7
Events
• An event is a subset of outcomes contained in
the sample space
– Can be simple (single event)
– Or compound (more than one outcome)
Say we are choosing sets of three marbles out of a box…
E1={RYB}
E2={YGR}
E3={RYB,YGR}
Compound event
8
Review of Set Theory
1. Intersection of events
A B
2. Union of events
A B
3. Mutually Exclusive Events
4. Complementary Event
A'
9
A more meaningful example
100 lb
RA
20 ft
RB
If the load must be placed at even 2 ft intervals, what is the
sample space of all pairs of RA and RB?
There are 10 pairs of reaction forces – solve the equilibrium
problem to get a set of general equations.
10
If the load must be placed at even 2 ft
intervals, what is the sample space of all
pairs of RA and RB?
RA
Sample Space
100 lb
100 lb
RA
20 ft
RB
RB
100 lb
In this case, we have a
deterministic load value, load
location varied discretely.
Now…what if the load could be 100 lb, 200 lb,
or 300 lb, and could be placed anywhere on the
beam?
11
RA
300 lb
RA
Sample Space
300 lb
Sample Space
200 lb
100 lb
100 lb
100 lb 200 lb 300 lb
RB
Sample space for load placement varied
continuously, load varied discretely.
RB
100 lb
300 lb
Sample space for load placement varied
continuously, load varied continuously
between 100 lb and 300 lb.
12
Event subspaces and set operations
A={RA>=100lb}
B={RB>=100lb}
RA
RA
300 lb
300 lb
A
100 lb
100 lb
B
RB
100 lb
300 lb
RB
100 lb
300 lb
13
A B
A B
RA
RA
300 lb
300 lb
A
100 lb
100 lb
B
100 lb
RB
300 lb
100 lb
300 lb
14
Counting Techniques
• Sometimes the sample space is small enough
to list all possibilities, or to visualize pictorially,
as we have been doing.
• More often than not we may have to apply
some general counting priniciples.
– Product rule
– Tree diagrams
– Permutations
– Combinations
15
Product Rule for k-tuples
• k-tuples are just ordered sets of k numbers,
with each number having nk possible values
• If we have 2 numbers, then we refer to it as an
ordered pair.
• Product rule says we can get the total number
of possible outcomes of a set of ordered ktuples by simply multiplying their nk possible
values together.
16
Product rule examples
• Think about a contractor example (found in
your packet)…
– We have ordered pairs (Plumber, Drywall); There
are 15 plumbing contractors and 5 drywall
contractors
– This means that our sample space consists of
• 15*5 = 75 possible outcomes
• What if we added a third contractor, with only
two options?
– Now there are 15*5*2 = 150 possible outcomes
17
Tree diagrams
• Can construct these to help visualize, may not
be practical for large sample spaces.
Job 1 – 1st generation
Job 2 – 2nd generation
Job 3 – 3rd generation
Count last generation
branches for total
number of sample
space possibilities.
18
Permutation
• So far we have looked at sampling with replacement (an element
can appear more than once), where the order of the k-tuple
matters
• What if we sample without replacement and order matters?
– The k-tuple is formed by selecting successively from this set without
replacement so that an element can appear in at most one of the k
positions.
– In this case, the number of possible values in the sample space is
defined by the equation below, where n is the total number of
elements in the set and k is the size of the grouping.
– Factorial notation: n!=n(n-1)(n-2)…(2)(1); 0!=1
n!
Pk ,n 
(n  k )!
19
Combination
• Sampling without replacement when order
does not matter.
• We are given n distinct objects and any
unordered subset of size k of the objects is
called a combination.
 n  Pk ,n
n!
  

 k  k! k!(n  k )!
20
Product Rule, Permutation, or
Combination???
1. You are looking on Travelocity for a flight to Tahiti
for Spring Break. There are three legs of the flight.
Each flight leg has 4 possible airlines to choose
from.
21
Product Rule, Permutation, or
Combination???
1. You are looking on Travelocity for a flight to Tahiti for
Spring Break. There are three legs of the flight.
Each flight leg has 4 possible airlines to choose from.
PRODUCT RULE: There are k
elements and each can be
selected from its own set of nk
possible values.
22
Product Rule, Permutation, or
Combination???
2. You are playing poker with your pals and each
player receives a hand of five cards out of the 52
card deck.
23
Product Rule, Permutation, or
Combination???
2. You are playing poker with your pals and each
player receives a hand of five cards out of the 52
card deck.
COMBINATION: Sample
without replacement, order
doesn’t matter
24
Product Rule, Permutation, or
Combination???
3. You and two of your most athletic friends decide
to complete in a relay triathlon. How many
different ways are there to order the three of
you?
25
Product Rule, Permutation, or
Combination???
3. You and two of your most athletic friends decide
to complete in a relay triathlon. How many
different ways are there to order the three of
you?
PERMUTATION: Sample without
replacement, order matters
26
Homework
Problems 1-5 in Probability Course Packet
27