Topics for Today
Introduction to Probability
Events
Probability Rules
Probability Distribution
Stat203
Fall 2011 – Week 4, Lecture 1
Page 1 of 28
Unlucky, Smartie-loving Dave
2 years ago Dave bought a laptop.
Within the first 12 months, (by warranty) he had
replaced:
 buetooth (x2),
 motherboard,
 heat sink,
 fans,
 ‘windows media button bar’ (x2),
 hard drive (x3) and
 company loyalty
A colleague who bought the same computer
had no such issues.
Stat203
Fall 2011 – Week 4, Lecture 1
Page 2 of 28
When a company doesn’t do what it says, it’s
expensive.
To limit risks a company needs to know the
_______ of a product failure or not being the
advertised weight.
The computer company needs to know the
chances (probability) that a laptop will ____ in
the first 1, 2 or 3 years. Because …
On a laptop
• Dell offers 1 year ‘free’ basic warranty.
• Upgrade the basic warranty to 3 years is an
extra $50
• Upgrade the basic warranty from 3 to 4 years
is an extra $260
Dell believes that the chances of costly repairs
increases with time and charge accordingly
Stat203
Fall 2011 – Week 4, Lecture 1
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What does this have to do with Probability?
______ = Probability
Probability is the __________ of an event
given all possible events
Probability = the number of times an event can
occur ÷ The total number of times that any
event can occur.
Probability Rule #1: Probability is a number
between _ (if it never ever happens) and
_ (if it always happens)
Stat203
Fall 2011 – Week 4, Lecture 1
Page 4 of 28
What’s an Event?
Possible events:
• ______________ OR computer doesn’t fail
For computer failure:
the ___________ of a computer failure = The
number of computers that fail ÷ The total
number of computers made
Stat203
Fall 2011 – Week 4, Lecture 1
Page 5 of 28
Possible events:
• _____________ OR don’t win lotto 649
the probability of a winning the lottery = The
number of winning numbers ÷ The total
number of possible numbers.
There is only 1 (one) winning set of 6
numbers.
There are 13,983,816 possible sets of 6
numbers that could come up.
(49 x 48 x 47 x 46 x 45 x 44 ) ÷ (6 x 5 x 4 x 3 x 2 x 1)
= 13,983,816
What’s the probability of winning?
1 ÷ 13,983,816 =
Stat203
Fall 2011 – Week 4, Lecture 1
Page 6 of 28
[ Aside on Notation:
P(A) is shorthand for the words “the probability
of the event A occurring”
From the prior examples we could write:
P(computer fails) = the probability a computer
fails
P(win 649) = the probability of winning lotto
649)
]
Stat203
Fall 2011 – Week 4, Lecture 1
Page 7 of 28
What about not winning Lotto 649?
P(Not win) = Nnot win ÷ Ntotal
There are 13,983,815 non-jackpot numbers
among the 13,983,816 _____________ of 6
numbers that could come up. So,
P(not win) = 13,983,815 ÷ 13,983,816
=0.99999992
In other words P(win) = 1- P(not win)
or P(win) = 1-P( winc )
winc is the _________of the event
[Another aside on notation … putting bars
above words (as on page 139 of the text) is
hard, so the superscript c will be used the
same way]
Stat203
Fall 2011 – Week 4, Lecture 1
Page 8 of 28
Probability Rule #2: the probability of an
event’s converse is 1 minus the probability of
the event.
What are converses?
Event
Win the lottery
Found Guilty
Randomly selected
person is a woman
Stat203
Fall 2011 – Week 4, Lecture 1
Converse
Don’t win the lottery
Not Found Guilty
Randomly selected
person is not a woman
Page 9 of 28
Example:
Selecting a person in the MTF study
In the MTF study, there are 6417 males and
6998 females.
If we ________ choose one row from the MTF
data set, the probability of it belonging to a a
females is
P(F) = Nfemale ÷ (Nfemale + Nmale)
P(F) = ____ ÷ (____+6417)
= 6,998 / 13,415
= ____
… or 52%
Stat203
Fall 2011 – Week 4, Lecture 1
Page 10 of 28
More Rules for Probability
Probability Rule #3: The additive rule means
that if you have 2 mutually exclusive events,
then the probability of one __ the other
occurring is the ___ of the probabilities of
either one.
In terms of rolling a dice, this means:
Define Events:
E1 = Dice shows 1
E2 = Dice is Even
What does P(E1 or E2) mean?
P(E1 or E2) = P(E1) + P(E2) = (1/6) + (3/6)
= 4/6
Mutually exclusive = both events cannot occur
Stat203
Fall 2011 – Week 4, Lecture 1
Page 11 of 28
Probability Rule #4: the multiplication rule
says that for two independent events, the
probability of one event ___ another is the
_______ of their probabilities
In terms of flipping two coins:
Define Events:
H1 = heads on first coin
H2 = heads on second coin
P(H1 and H2) = P(H1) x P(H2) = (1/2)(1/2)
=¼
Independent events are such that one event
doesn’t influence the probability of the other
Events for 2 coins = independent
Events for 2 friends = not independent
Stat203
Fall 2011 – Week 4, Lecture 1
Page 12 of 28
Example: Cancer Treatment
New hope for kids with leukemia as new drug
boosts survival rates
New drug sends survival rate soaring from 30% to 87%, B.C.led study says
BY PAMELA FAYERMAN, VANCOUVER SUN
http://www.vancouversun.com/health/hope+kids+with+leukemia+drug+boosts+survival+rates/2094516/story.html
Type of cancer:
Philadelphia chromosome-positive acute
lymphoblastic leukemia (Ph+ALL)
Only about six children and 90 adults get it
each year in Canada. The Phase 2 trial was
conducted at 20 cancer centres, mostly in the
U.S. (only two in Canada, Halifax and
Vancouver).
Stat203
Fall 2011 – Week 4, Lecture 1
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N = 157 patients
“aged one to 21 years, enrolled in the trial from 2002 to
2006. They were divided into comparison groups and
those who got the imatinib for the longest duration (280
continuous days, plus their intensive chemotherapy
regimen), had the best outcomes.”
Individuals?
Method of Capturing data?
Variables?
Stat203
Fall 2011 – Week 4, Lecture 1
Page 14 of 28
Let’s define some events:
• SnD = survive beyond 3 years no drug
• SwD = survive beyond 3 years with drug
What’s the converse for each of these events?
From the paper, they found
• P(SnD) = .30
• P(SwD) = .87
Stat203
Fall 2011 – Week 4, Lecture 1
Page 15 of 28
If two people have this type of Cancer and are
taking the drug what is the probability that
both survive for three years?
Are these two events independent? Why:
So, P(SwD and SwD) = P(SwD)*P(SwD)
= .87 * .87
= .76
What about the overall result? Can we really
say that P(SwD) > P(SnD) based on N = 157?
If we look at a bigger sample we might have a
better idea.
Stat203
Fall 2011 – Week 4, Lecture 1
Page 16 of 28
The Probability Distribution
The probability distribution is very similar to
the __________________ distribution or
_______ frequency distribution.
But: remember that the frequency distribution
related to the _______________ frequencies
in our dataset
The Probability distribution relates to the
___________ probability of all possible
observations in our dataset.
Stat203
Fall 2011 – Week 4, Lecture 1
Page 17 of 28
Events As coin flips (again)
Outcomes from tossing two coins:
• HH, TH, HT, TT
So:
P(2H) = ¼
P(1H) = 2/4
P(0H) = 1/4
This is the ________________________ of
the number of heads from 2 coin flips.
(we know coins [and dice and cards]pretty well so can
figure out the probability distribution explicitly … you
can’t do this for most events)
Stat203
Fall 2011 – Week 4, Lecture 1
Page 18 of 28
The probability distribution tells us the
___________ of _________ any event
Here is same the probability distribution of the
number of heads in two coin tosses shown
graphically:
Stat203
Fall 2011 – Week 4, Lecture 1
Page 19 of 28
When we toss the two coins 4 times, and look
at the _________ distribution, it probably won’t
look like the ___________ distribution.
Do 4 tosses of 2 coins and count the number
of heads:
Coin #1
Coin #2
Stat203
Fall 2011 – Week 4, Lecture 1
# Heads
Page 20 of 28
From this sample, we can count the frequency
of each possible number of heads:
# heads
Frequency
Relative
Frequency
0
1
2
Does this match the probability distribution we
saw earlier?
Stat203
Fall 2011 – Week 4, Lecture 1
Page 21 of 28
So making 4 tosses, is just like taking a
______ of 4 from a larger population of tosses
… there is ___________ from sample to
sample.
Just like the Smarties example earlier.
Stat203
Fall 2011 – Week 4, Lecture 1
Page 22 of 28
Key to Probability Distribution
Probability distribution = ___________
Frequency distribution = _________ (observed
from data)
We need a lot of data to try to figure out what
the theoretical probabilities are.
Stat203
Fall 2011 – Week 4, Lecture 1
Page 23 of 28
Example: The Bent Quarter
What’s the probability distribution of Heads for
a bent quarter?
Let’s start off by doing the frequency and
relative frequency for 10 tosses:
Stat203
Fall 2011 – Week 4, Lecture 1
Page 24 of 28
H?
Total # heads
So the relative frequency for Heads from the
bent coin is:
What’s our best guess at the probability
distribution for the H event?
Stat203
Fall 2011 – Week 4, Lecture 1
Page 25 of 28
A probability distribution is:
a) A Theoretical thing that we get from the
sample
b) The underlying pattern to the histogram (the
real curved line)
d) The probabilities of events from the
population
Stat203
Fall 2011 – Week 4, Lecture 1
Page 26 of 28
Today’s Topics
Introduction to Probability
- Chance
Rules
- Rule 1: always between 0 and 1
- Rule 2: converse
- Rule 3: addition
- Rule 4: multiplication
Probability Distribution
- Theoretical version of the relative frequency
distribution
- Is approximated by the relative frequency
distribution
- Don’t always know it
- Is represented by a histogram
Stat203
Fall 2011 – Week 4, Lecture 1
Page 27 of 28
Reading for next lecture
Chapter 5 – The Normal Curve
Stat203
Fall 2011 – Week 4, Lecture 1
Page 28 of 28