MAT199: Math Alive
Probability and Statistics
Ian Griffiths
Mathematical Institute, University of Oxford,
Department of Mathematics, Princeton University
Probability and Statistics
• Probability arises in many forms of everyday life:
e.g.,
• There is a one in six chance of rolling a 6 on a die.
Probability and Statistics
• Probability arises in many forms of everyday life:
e.g.,
• There is a one in six chance of rolling a 6 on a die.
• There is a 30% chance of rain today.
Probability and Statistics
• Probability arises in many forms of everyday life:
e.g.,
• There is a one in six chance of rolling a 6 on a die.
• There is a 30% chance of rain today.
• There is a 1% chance that Princeton will win March Madness in the next ten
years.
Disease testing
• While at the doctors’ surgery, the doctor decides to do a routine test for a very rare
disease which affects only 1 in 10,000 people.
• The test is 99% accurate.
• The test comes back positive. What should the doctor say?
Disease testing
• While at the doctors’ surgery, the doctor decides to do a routine test for a very rare
disease which affects only 1 in 10,000 people.
• The test is 99% accurate.
• The test comes back positive. What should the doctor say?
(a) “It’s bad news I’m afraid…”
Disease testing
• While at the doctors’ surgery, the doctor decides to do a routine test for a very rare
disease which affects only 1 in 10,000 people.
• The test is 99% accurate.
• The test comes back positive. What should the doctor say?
(a) “It’s bad news I’m afraid…”
(b) “Don’t worry, it’s still very unlikely you have the disease.”
Disease testing
• While at the doctors’ surgery, the doctor decides to do a routine test for a very rare
disease which affects only 1 in 10,000 people.
• The test is 99% accurate.
• The test comes back positive. What should the doctor say?
(a) “It’s bad news I’m afraid…”
(b) “Don’t worry, it’s still very unlikely you have the disease.”
He should say (b). In fact, the patient still has less than 1% chance of having the disease.
Cups and prizes
• A prize is placed under one cup. What is the probability that you guess the correct cup?
1
2
• The probability of guessing the correct cup is one in three.
3
Cups and prizes
• A prize is placed under one cup. What is the probability that you guess the correct cup?
1
2
3
• The probability of guessing the correct cup is one in three.
• You choose a cup. I then reveal that the prize is not under one of the other cups
and offer you a chance to switch. Should you stick or switch?
• If you stick then your chance of winning is still one in three. But what are your
chances if you switch…?
Cups and prizes
• Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is
under cup 2 or cup 3.)
3
2
1
Cups and prizes
• Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is
under cup 2 or cup 3.)
3
2
1
• If you choose to switch then there are three possible outcomes:
1) You choose cup 1 (containing the prize). I uncover one of the other empty cups.
You switch to the remaining covered cup. You lose.
Cups and prizes
• Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is
under cup 2 or cup 3.)
3
2
1
• If you choose to switch then there are three possible outcomes:
1) You choose cup 1 (containing the prize). I uncover one of the other empty cups.
You switch to the remaining covered cup. You lose.
2) You choose cup 2. I uncover the empty cup (cup 3). You switch to the remaining
covered cup (cup 1). You win.
Cups and prizes
• Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is
under cup 2 or cup 3.)
3
2
1
• If you choose to switch then there are three possible outcomes:
1) You choose cup 1 (containing the prize). I uncover one of the other empty cups.
You switch to the remaining covered cup. You lose.
2) You choose cup 2. I uncover the empty cup (cup 3). You switch to the remaining
covered cup (cup 1). You win.
3) You choose cup 3. I uncover the empty cup (cup 2). You switch to the remaining
covered cup (cup 1). You win.
Cups and prizes
• Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is
under cup 2 or cup 3.)
3
2
1
• If you choose to switch then there are three possible outcomes:
1) You choose cup 1 (containing the prize). I uncover one of the other empty cups.
You switch to the remaining covered cup. You lose.
2) You choose cup 2. I uncover the empty cup (cup 3). You switch to the remaining
covered cup (cup 1). You win.
3) You choose cup 3. I uncover the empty cup (cup 2). You switch to the remaining
covered cup (cup 1). You win.
So if you switch then you win two times out of three. This is twice as good a chance
of winning than if you had stayed with your original choice.
Bayes’ theorem
Thomas Bayes
1701–1761
Summary of lecture 13
• Probability of an event A occurring is denoted by p(A)
• We use three rules for calculating probabilities:
p(A or B) = p(A) + p(B)
(1)
p(A and B) = p(A) x p(B)
(2)
p(A given B) = p(A and B)
p(B)
(3)
This is called
Bayes’ theorem
What is the probability of two people in a room
sharing a birthday?
What is the probability of two people in a room
sharing a birthday?
Probability of sharing a birthday
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
Number of people in the room
50
What is the probability of two people in a room
sharing a birthday?
Probability of sharing a birthday
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
Number of people in the room
• You need only 22 people in a room for the probability of two people sharing a
birthday to be greater than 0.5.
Statistics
• “I can prove anything with statistics except the truth.”
George Canning.
Statistics
• “I can prove anything with statistics except the truth.”
George Canning.
Unjustified accuracy
• 87.452522% of all statistics claim a precision of results that are not backed up by
the method used.
Statistics
• “I can prove anything with statistics except the truth.”
George Canning.
Unjustified accuracy
• 87.452522% of all statistics claim a precision of results that are not backed up by
the method used.
Manipulating averages
• A statistician died crossing a river that was, on average, six inches deep.
Statistics
• “I can prove anything with statistics except the truth.”
George Canning.
Unjustified accuracy
• 87.452522% of all statistics claim a precision of results that are not backed up by
the method used.
Manipulating averages
• A statistician died crossing a river that was, on average, six inches deep.
Plain wrong
• Math illiteracy affects 8 out of every 5 people.
• If there is a 50/50 chance of something going wrong then 9 times out of 10 it will.
Summary of lecture 14
• The expected value for an event,
E = (probability of event 1 happening) x (value for event 1)
+ (probability of event 2 happening) x (value for event 2)
+… + (probability of event N happening) x (value for event N)
• The variance of an event,
Variance = (probability of event 1 happening) x (value for event 1 – Expectation value, E)2
+ (probability of event 2 happening) x (value for event 2 – Expectation value, E) 2
+… + (probability of event N happening) x (value for event N – Expectation value, E) 2
• The standard deviation of an event =
variance
Pascal’s triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
…
What happens if we take more points?
n =10
What happens if we take more points?
n =10
n =20
What happens if we take more points?
n =10
n =20
n =100
What happens if we take more points?
n =10
n =20
n =100
This is called the
binomial distribution
Analysing the final curve
• We can scale the distribution so that the sum of all the possible outcomes = 1 and
the values correspond to probabilities.
Analysing the final curve
p
n = 100
r
• We can scale the distribution so that the sum of all the possible outcomes = 1 and
the values correspond to probabilities.
• The result is approximated by the formula:
Analysing the final curve
p
n = 100
r
• We can scale the distribution so that the sum of all the possible outcomes = 1 and
the values correspond to probabilities.
This is the
mean value
• The result is approximated by the formula:
Analysing the final curve
p
n = 100
r
• We can scale the distribution so that the sum of all the possible outcomes = 1 and
the values correspond to probabilities.
This is the
mean value
• The result is approximated by the formula:
This is related
to the spread of
the data
The normal distribution
The normal distribution
• In general we write:
The normal distribution
• In general we write:
The mean
value
The
standard
deviation
The normal distribution
• In general we write:
The mean
value
The
standard
deviation
• This is called the normal distribution.
The normal distribution
• The normal distribution is also known as a bell curve.
Summary of lecture 15
• The normal distribution:
0.08
Tossing a coin 100 times. This
shows the probability, p, of
getting a head a total of r times.
0.06
p
0.04
0.02
20
40
60
r
80
100
Summary of lecture 15
• The normal distribution:
0.08
Tossing a coin 100 times. This
shows the probability, p, of
getting a head a total of r times.
0.06
p
0.04
0.02
20
40
60
r
80
100
Summary of lecture 15
• The normal distribution:
0.08
Tossing a coin 100 times. This
shows the probability, p, of
getting a head a total of r times.
0.06
p
In this example the mean is m =
50
and the standard deviation is
σ = 10.
0.04
0.02
20
40
60
r
80
100
The mean
value
The
standard
deviation
Summary of lecture 15
• The normal distribution:
0.08
Tossing a coin 100 times. This
shows the probability, p, of
getting a head a total of r times.
0.06
p
In this example the mean is m =
50
and the standard deviation is
σ = 10.
0.04
0.02
20
40
60
r
80
100
The mean
value
The
standard
deviation
• This kind of distribution arises in many situations, e.g. distribution of heights of
people in the U.S.
The Griffiths University
The Griffiths University
• Here are some statistics for the 2012 intake of students at the Griffiths University:
Success rate for males who applied to the Griffiths University: 17%
Success rate for females who applied to the Griffiths University: 82%
The Griffiths University
• Here are some statistics for the 2012 intake of students at the Griffiths University:
Success rate for males who applied to the Griffiths University: 17%
Success rate for females who applied to the Griffiths University: 82%
• Is the Griffiths University a fair University?
(a) Yes.
(b) No.
(c) Don’t know.
Representing data
Good forms of data representation
Representing data
Good forms of data representation
• This is a visualization of the military campaign of Bonaparte against Russia. The
band shows the route taken by Napoleon’s army. The thickness of the band is
proportional to the number of surviving troops.
Representing data
Good forms of data representation
Representing data
Good forms of data representation
• This shows the air pollution in Los Angeles at different times of the day.
Representing data
Bad forms of data representation
Representing data
Bad forms of data representation
• Where is the best place to sit in the Math Alive lectures?
Midterm grades for students who sit at
the front of the lecture theatre.
Midterm grades for students who sit
at the back of the lecture theatre.
Representing data
Bad forms of data representation – exploiting viewing angles
• We can also use perspective to skew data…
Representing data
Bad forms of data representation – exploiting viewing angles
• We can also use perspective to skew data…
• This is a fairer way of representing the data:
Representing data
Bad forms of data representation – exploiting viewing angles
Representing data
Bad forms of data representation
• Here we show the Nobel prizes won
by various countries. Can you see
why this representation might be
unfair?