Download Probability

Probability - 1
Probability statements are about likelihood,
NOT determinism
Example: You can’t say there is a 100%
chance of rain (no possibility of not
having rain – i.e., a certainty, not
a probability)
Probability - 2
• Flip a coin. Will it be heads or tails?
• The outcome of a single event is random,
or unpredictable.
• What if we flip a coin 10 times? How many
will be heads and how many will be tails?
• Over time, patterns emerge from seemingly
random events. These allow us to make
probability statements.
Heads or tails? – The Relative
Frequency Concept of Probability
• A computer simulation of 10,000 coin flips
yields 5040 heads. What is the relative
frequency of heads?
• 5040 / 10,000 = .5040
Each of the tests is the result of a sample of fair
coin tosses.
Sample outcomes vary.
• Different samples produce different results.
True, but the law of large numbers tells us
that the greater the number of repetitions the
closer the outcomes come to the true
probability, here .5.
A single event may be unpredictable but the
relative frequency of these events is lawful
over an infinite number of trials\repetitions.
Random Variables
• "X" denotes a random variable. It is the outcome
of a sample of trials.
• “X,” some event, is unpredictable in the short run
but lawful over the long run.
• This “Randomness” is not necessarily
unpredictable. Over the long run X becomes
probabilistically predictable.
• We can never observe the "real" probability, since
the "true" probability is a concept based on an
infinite number of repetitions/trials. It is an
"idealized" version of events.
To figure the odds of some event
occurring you need 2 pieces of
information:
1. A list of all the possibilities – all the
possible outcomes (sample space)
2. The number of ways to get the
outcome of interest (relative to the
number of possible outcomes).
How Many Ways can Two Dice Fall?
Let’s say the dice are different colors (helps us
keep track).
The White Dice could come out as:
We know how to figure out probabilities here, but
What about the other dice?
• When the white die shows
possible outcomes.
, there are six
• When the white die shows
more possible outcomes.
, there are six
• We then just do that for all six possible
outcomes on the white die
• What is the probability of rolling numbers that
sum to 4?
• What do we need to know?
– All Possible Outcomes from rolling two dice
• (36--Check Previous Slide)
– How many outcomes would add up to 4?
Our Probability is 3/36 = .08333
Probability = Frequency of Occurrence
Total # outcomes
Frequency of occurrence = # of ways this
one event could happen
Total # outcomes = # ways all the
possible events could happen
Probability of a 7 is 6 ways out of 36
possibilities
p=.166
Expected Frequency of the Sum of 2 Dice
p = .167
6
.139
.139
5
F
.111
4
.111
.083
3
.083
.056
2
.056
.027
.027
1
2
3
4
5
6
7
8
9
Sum of two dice
10
11
12
You Are Flipping 4 Coins
• What are the chances of getting 2 heads and
2 tails?
• Find the possible number of outcomes…
Histogram of Outcomes: X = Number of Heads
TTTT
------X=0
HTTT
THTT
TTHT
TTTH
------X=1
OUTCOME
0
PROB
.0625
HTTH
HTHT
THTH
HHTT
THHT
TTHH
------X=2
1
.25
2
.375
HHHT
HHTH
HTHH
THHH
------X=3
HHHH
------X=4
3
.25
4
.0625
Divide the number of 2-head/2-tail outcomes by the number
Of possible outcomes: 6/16 = .375
“THE
LAW OF LARGE NUMBERS”:

If we observe a large number of outcomes of a
random variable and then calculate the mean of this
distribution, this random variable will increasingly come
close to the true mean of the distribution.
• The relative frequency increasingly comes to
center on the true probability and eventually
becomes stable.

Over many repetitions the sample mean is an
unbiased estimator of the population mean, for coin
tosses  .5
Much of statistics is based of establishing the
odds, the likelihood, that a single event or small set
of events could have occurred by chance.
EXAMPLE: H H H H H -- 5 Heads in 5 tosses -- is
possible but rather unlikely. Will happen on average
with a probability of
 .5 * .5 * .5 * .5 * .5 = .03.
So, if 100 people toss 5 coins each, 3
out of 100 will get 5 heads or 5 tails in a
row. Not so odd -- well within the
realm of possibility – in fact it will
routinely happen.
To be expected. So much of what we
think is strange, odd, miraculous is
really predictable --the law of
randomness at work.
The odds of events occurring -what outcomes chance alone would
produce.
How different are the outcomes
you obtain from a given sample
compared to what results you could
get solely due to chance?
The “Expected Value” of a variable
is the total value of the scores on
that variable times the probability
of occurrence. If there is a contest
among four people in which one
person will win $8 and three people
will win $0 the expected value for
each person is:
Expected Value = $8 x .25 = $2
Note: In this example no one will
receive the “expected value.”
This is one major reason why progress
in the war on cancer is so slow. If a
drug company can charge the same price
for a “small improvement” cancer
treatment (e.g., extends life 3 months) as
a “curative” treatment and the “small
improve” treatment is more probable,
then the drug company’s incentive (i.e.,
“expected value”) is greater for doing
research trials on the “small
improvement” treatment than the
“curative” treatment.
In a similar vein, this is why there is so
little research on promising “alternative
cancer treatments” (e.g., Coley’s
Toxins): since the drugs used in the
treatment are “common” (i.e., routinely
available in stores), the treatment can’t
be patented. Thus, the drug company
can’t profit to nearly the extent it can
with a patented, but potentially less
successful, treatment. Health wise, what
is in your “best interest” is NOT
necessarily in the drug company’s best
interest.
Bayes’s Theorem – Another
Concept of Probability
Bayes’s theorem of probability takes into
account the totality of the circumstances
surrounding an event. For example, if
1,000,000 lottery tickets are sold, then,
using the relative frequency concept of
probability, you have a 1/1,000,000 chance
of winning.
Bayes’s Theorem - 2
However, if the evening news has a high probability
of accurate reporting, and they report that your
number has been selected, then, by Bayes’s
theorem, the probability of you winning is much,
much greater than 1/1,000,000. Since we typically
can’t estimate the probability of the circumstances
surrounding an event, we won’t use Bayes’s
theorem.
The Central Limit Theorem - 1
• If we take repeated samples from a
population, the sample means will be
(approximately) normally distributed.
– The mean of the “sampling distribution” will
equal the true population mean.
– The “standard error” (the standard deviation of
the sample distribution) is analogous to the
standard deviation.
The Central Limit Theorem - 2
• A “sampling distribution” of a statistic tells us what values
the statistic takes in repeated samples from the same
population and how often it takes them.
The Central Limit Theorem - 3
• We use the statistical properties of a distribution
of many samples to see how confident we are that
a sample statistic is close to the population
parameter. We can compute a confidence interval
around a sample mean or a proportion:
– We can pick how confident we want to be
– Usually choose 95%, or two standard errors –
Remember, with a normal distribution 95% of
the cases are within two standard deviations of
the mean.
Central Limit Theorem - 4
Let’s say I do a study of political
contributions. For sample #1 I randomly
select 1,000 individuals and compute the
mean dollar contribution per person. If I
repeat this process 399 times (i.e., a total of
400 samples of 1,000 randomly selected
individuals each), the distribution of the 400
sample means will be approximately
normal.
Central Limit Theorem - 5
Thus, if the mean of the sample means is $25
per person (i.e., I add up all 400 sample
means, divide the total by 400 and the
average is $25) and the standard deviation
of these sample means is $5, approximately
95% of these sample means will be between
$15 and $35. This is the central limit
theorem. It will be of critical importance!
Two Principles of Tests of
Statistical Significance
1. How great is the relationship between X
and Y?
2. How many observations are used in
estimating the relationship between X and
Y?
The Null Hypothesis
The null hypothesis is that the independent
variable has no influence on the dependent
variable. For example, the party affiliation
of the president is unrelated to the policies
they pursue.
Errors in Hypothesis Testing
Type 1 Error: reject the null hypothesis when
the null hypothesis is true.
Type 2 Error: retain the null hypothesis when
the null hypothesis is false.
For us, a Type 1 error is of greater importance
than a Type 2 error.
Decision Rule
If the results are statistically significant at the .05
level it means the following: (1) we will reject the
null hypothesis 100% of the time; (2) 95% of the
time we will have made the correct decision
because the null hypothesis will be false 95% of
the time; (3) 5% of the time we will have
committed a type I error because we will have
rejected the null hypothesis when the null
hypothesis is true; (4) we will never know for
certain if the null hypothesis is false.
Visualizing Hypothesis Testing
Probability – Just for Fun
What is the logic of the following statement
by famous Las Vegas casino owner Bob
Stupak:
“I only make money when you win!”
Just for Fun!
Can you explain the logic of the following
statement by former Las Vegas casino
owner Bob Stupak: “I only make money
when you win!”