Statistics
Introduction
• The study of probability is often deceptive:
• on the surface, it seems close to everyday
experience and intuition seems enough to find
answers to problems.
• Terms such as "randomness," "chance," and
so forth are used by laypeople such as the
media, to justify one action over another.
• The mathematical notion of probability is a
different case.
• Terms are well-defined, rules formulated and
proven, reason preferred over intuition.
Question One
• "A truth serum given to a suspect is known to
be 90% reliable when the person is guilty. If
you are guilty and you are given the serum,
what is the probability that you will go free?”
• From common sense, you would know the
answer is 10%.
Question Two
• "A truth serum given to a suspect is known to
be 90% reliable when the person is guilty and
99% reliable when the person is innocent.
• If the suspect was selected from a group of
suspects of which only 5% have ever
committed a crime, and the serum indicates
that he is guilty, what is the probability that he
is innocent?”
This needs some careful thinking and
consideration but can be solved.
• "A truth serum given to a suspect is known to
be 90% reliable when the person is guilty and
99% reliable when the person is innocent.
• If the suspect was selected from a group of
suspects of which only 5% have ever
committed a crime, and the serum indicates
that he is guilty, what is the probability that he
is innocent?”
This version cannot be solved!
• A truth serum given to a suspect is known to
be 90% reliable when the person is guilty and
99% reliable when the person is innocent.
• If the suspect was selected from a group of
suspects of which only a few of which have
ever committed a crime, and the serum
indicates that he is guilty, what is the
probability that he is innocent?
There is a contradiction in the question
• A truth serum given to a suspect is known to
be 90% reliable when the person is guilty and
99% reliable when the person is innocent.
• If the suspect was selected from a group of
suspects of which only 5% have ever
committed a crime. Furthermore, 1% of the
guilty are judged innocent by the serum. The
serum indicates that he is guilty, what is the
probability that he is innocent?
One part is irrelevant to answering the
question
• A truth serum given to a suspect is known to
be 90% reliable when the person is guilty and
99% reliable when the person is innocent.
25% of the suspects are scared to die.
• If the suspect was selected from a group of
suspects of which only 5% have ever
committed a crime, and the serum indicates
that he is guilty, what is the probability that he
is innocent?
Subtle irrelevancy
• A truth serum given to a suspect is known to be
90% reliable when the person is guilty and 99%
reliable when the person is innocent. 25% of the
suspects are scared to die.
• If the suspect was selected from a group of
suspects of which only 5% have ever committed a
crime and only 1% have ever been found guilty,
and the serum indicates that he is guilty, what is
the probability that he is innocent?
Reword (in your head) to get the
correct information
• "A truth serum given to a suspect is known to be 90%
reliable when the person is guilty and 99% reliable
when the person is innocent. This means that 90% of
the guilty ones are judged guilty by the serum, and
10% of the guilty ones are judged innocent. Also, this
means that 99% of the innocent ones are judged
innocent, and 1% of the innocent ones are judged
guilty. The suspect was selected from a group of
suspects of which only 5% have ever committed a
crime--this means that 5% are guilty. If the serum
indicates that the suspect is guilty, what is the
probability that he is innocent?"
Question Three
• Based on a conversation with somebody you
meet for the first time you discover that this
person has at least one son.
You subsequently discover that this person
has two children.
What is the probability that the other child is a
boy?
Answer
First child
B
B
G
G
1
3
Second child
B
G
B
G
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
• A. 25%
• B 50%
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct? The confusion results from two
• A. 25% sources
• B 50% 1) There are percentage signs in
• C 60% the answers
• D 25% 2) The fact that 50% of the
answers are “25%” and 25% of
the answers are “50%”
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
The question does not say that
• A. 25%
the answer must among the
• B 50%
listed options. It only asks what
• C 60%
is the probability that you will
• D 25%
be correct if you answer at
random.
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
The question instructs to only
• A. 25%
choose one single answer out
• B 50%
of four. And assume a uniform
• C 60%
distribution, since that is most
• D 25%
likely intended, then each
answer has a chance of 25% to
become chosen.
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
So the correct answer should
• A. 25%
be: 25%.
• B 50%
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
This computes to answer A
• A. 25%
being correct, as well as answer
• B 50%
D. Could that be?
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Yes, it can. The question does
• A. 25%
not reveal how many of the
• B 50%
four given answers are correct,
• C 60%
but since there is one to be
• D 25%
picked, assume that at least
one of the four answers is
correct.
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Let's call answer A + answer D
• A. 25%
the correct answer pair.
• B 50%
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Now, there are two possible
• A. 25%
choices (A or D) that result in
• B 50%
50% of the correct answer (A
• C 60%
and D). Secondly, there is 50%
• D 25%
chance of picking one (A or D)
of two (A and D) out of four (A
to D).
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
So whether answer A or answer
• A. 25%
D is chosen, in either case the
• B 50%
probability of being correct
• C 60%
(50% × 50%) is 25%, which
• D 25%
evaluates true.
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Thus, yes, the question has 2
• A. 25%
correct answers.
• B 50%
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Do you agree with this logic?
• A. 25%
• B 50%
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
The question starts:
• A. 25%
"If you choose an answer to this
• B 50%
question at random,
• C 60%
• D 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
However it does not then
• A. 25%
continue:
• B 50%
"what is the probability that the
• C 60%
answer chosen will be the
• D 25%
probability of choosing that
answer?”
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
It instead says:
• A. 25%
"what is the probability that
• B 50%
you will be correct?"
• C 60%
And then doesn't define
• D 25%
correct.
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
However, lets presume that one
• A. 25%
of the following answers can be
• B 50%
chosen:
• C 60%
a, b, c or d
• D 25%
The probability of choosing
each answer:
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
a - 25%
• A. 25%
b - 25%
• B 50%
c - 25%
• C 60%
d - 25%
• D 25%
25% - 50%
50% - 25%
60% - 25%
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Being correct for "what is the probability
• A. 25% that you will be correct?" if there is one
• B 50% correct answer (although as covered
above the question doesn't define
• C 60%
correct or specify how many answers are
• D 25% correct). This produces the same answer
as the question "is the answer you
choose the probability of choosing that
answer?”
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
a – yes
• A. 25% b - no
• B 50% c - no
d - yes
• C 60%
25% - yes
• D 25% 50% - no
60% - no
Question Four
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Could this be the real question in the
• A. 25% question. "what is the probability that
• B 50% the answer chosen will be the
probability of choosing that answer?":
• C 60%
• D 25% In which case, ‘b’ is correct!
Does this help?
• If you choose an answer to this question at
random, what is the chance you will be
correct?
• A) 250
• B) 500
• C) 600
• D) 250
What if we start with this?
• If you choose an answer to this question at
random, what is the chance you will be
correct?
1. A
2. B
3. C
4. D
What if we start with this?
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Provided that one of the
1. A
four option is correct
2. B
(assumption), the
3. C
probability will be 25%
4. D
that you are correct.
What if we start with this?
• If you choose an answer to this question at
random, what is the chance you will be
correct?
Since you are picking up a
random answer (don't bother
1. A
about
logic
at
all),
the
correct
2. B
answer to the question "what is
3. C
the probability that you will be
4. D
correct" is 25%. Do not bother
about the options, which is
misleading.
What if we start with this?
• If you choose an answer to this question at
random, what is the chance you will be
correct?
But if there are two choices of
25% which is technically wrong
1. A
because
every
choice
must
be
2. B
different from the another.
3. C
4. D
What if we start with this?
• If you choose an answer to this question at
random,
what isof
the
chance
you will
be
Chances
each
selection
at random
correct?are 1/4 or 25%. That means 25%
would select choice A at random, 25%
1. A
choice
B
at
random
and
so
on.
Since
2. B
we know choice A and D are the
3. C
correct answer (they are repeated
4. D
which is wrong), that leads us to 50%
correct answer if people make
random choices.
What if we start with this?
• If you choose an answer to this question at
random, what is the chance you will be
correct?
1. A
2. B
So what is the probability that you
will be correct is 50%
3. C
4. D
Each of the following statements is
either true or false. Which of them
are true and which are false?
1.
2.
3.
4.
5.
All of these sentences are false.
Exactly 1 of these sentences is true.
Exactly 2 of these sentences are true.
Exactly 3 of these sentences are true.
Exactly 4 of these sentences are true.
Sample question
• Imagine you have been tested in a large-scale
screening programme for a disease known to
affect one person in a hundred. The test is 90%
accurate, and you test positive. What is the
probability that you have the disease?
True or False
The squares marked A and B
are the same shade of gray,
yet they appear different
True!
Question Two
• http://www.agenarisk.com/resources/probabi
lity_puzzles/Making_sense_of_probability.ht
ml
• http://web.mit.edu/persci/gaz/gazteaching/index.html
Monty Hall Problem
• http://www.grandillusions.com/simulator/montysim.htm
http://www.math.ucsd.edu/~crypto/
Monty/monty.html
Birthday Problem
• http://wwwstat.stanford.edu/~susan/surprise/Birthday.ht
ml
• Suppose a crime has been committed and that
the criminal has left some physical evidence,
such as some of their blood at the scene.
• Suppose the blood type is such that only 1 in
every 1000 people has the matching type.
•
http://www.agenarisk.com/resources/probability_puzzles/prosecutor.shtml
• A suspect, let's call him Fred, who matches the
blood type is put on trial. The prosecutor
claims that the probability that an innocent
person has the matching blood type is 1 in a
1000 (that's a probability of 0.001).
• Fred has the matching blood type and
therefore the probability that Fred is innocent
is just 1 in a 1000.
• But the prosecutor’s assertion, which sounds
convincing and could easily sway a jury, is
wrong.
• A fair dice with 6 sides is rolled a certain
number of times and the number 1-6 is
recorded each time it is rolled. Which of the
following sequences (exact order) has the
greatest probability of occurring?
A) 12345
B) 654321
C) 2222
D) 241523
• A die is rolled, find the probability that an
even number is obtained.
What were your assumptions?
• Two coins are tossed, find the probability that
two heads are obtained.
What were your assumptions?
Did you make more than one assumption?
Answer the question completely!
• Two dice are rolled, find the probability that
the sum is
a) equal to 1
b) equal to 4
c) less than 13
And again!
• A die is rolled and a coin is tossed, find the
probability that the die shows an odd number
and the coin shows a head.