Daniel Kahneman wins 2002 Nobel Prize

for economics

His work showed that people routinely fail

all sorts of tests of economic rationality.

Most of the illustrations are statistical in character.

Sad note…. Most of Kahneman’s work

was done with Amos Tversky,who died a few years ago.

• The Nobel Prize goes only to the living.

Here’s an example of their work.

There are two hospitals in a medium-size city. At the smaller hospital, there are about 15 births per day. At the larger hospital, there are about 45 births per day. We are counting the number of days, out of 365 in a calendar year, in

which there are 60% or more boy babies.

Choose one of the following:

At the larger hospital there will be more days on which 60% or more of the babies are boys. Each of the two hospitals will have about the same number of days on which 60% or more of the babies are boys. At the smaller hospital there will be more days on which 60% or more of the babies are boys.

The answer:

The smaller hospital will have more days on which 60% or more of the

babies are boys.

Suppose that X is the random number of boys in the hospital

with 15 births per day. Then X is binomial with n = 15 and p = 0.5

Note that E X = np = 7.5.

The condition “60% or more boys” means X 9.

15 0.5 0.5

Note that SD(X) =

1.94

The target of 9 is

standard deviations above the mean.

9 7.50.77

1.94

Let Y be the number of boys at the larger hospital.

Y is binomial with n = 45 and p = 0.5

Note E Y = 22.5 andSD(Y) =

45 0.5 0.5 3.35

The target “60% or more boys” is 27. This is

standard deviations above the mean.

27 22.51.34

3.35

Some exact calculations:

P[ X 9 ] = 0.3036

P[ Y 27 ] = 0.1163

The commentary?

People have great difficulty in assessing the role of sample size in probabilistic calculations.

Here’s another example.

Suppose that you are doing a binomial experiment in which the outcomes can be described as RED and BLUE.

Say that p = probability of RED outcome = .

Which sequence of ten trials is more likely…

G: R R R R R R R R R RH: R B R R B B R R R B

We can do the math, but for now just guess.

4

5

It turns out that sequence G is more likely.

Here are the exact probabilities:

For G, the probability is 0.1074.

For H, the probability is

0.0881.

104

5

6 410 4 1

4 5 5

It seems that people really want random outcomes to look random.

….and one more…

There are two programs in a high school. Boys are a majority (65%) in program A.Boys are a minority (45%) in program B. There are equal numbers of classes in the two

programs. You enter a class at random and observe that 55%

of the students are boys. What is your best guess – does the class belong to program A or program B?

We know P(A) = P(B) = 0.50.

Assume for convenience that the class has 20 students. Thus 55% boys means that 11 of the 20 are boys.

Find then P(11 boys | A) = 0.1158.

Also P(11 boys | B) = 0.1185.

Note that P(11 boys | B) > P(11 boys | A).

11 9200.65 0.35

11

11 9200.45 0.55

11

Read carefully, as 0.1158 0.1185.

We can use Bayes’ theorem to get P(A | 11 boys). It is

=

= <

P P 11 boys |

P P 11 boys | P P 11 boys |

A A

A A B B

1P 11 boys |

21 1

P 11 boys | P 11 boys |2 2

A

A B

P 11 boys |

P 11 boys | P 11 boys |

A

A B1

2

The observed 55% boys is a little bit closer

to the program with 45% boys (B)

thanto the program with 65% boys (A).

Why?

The binomial standard deviation with p = 0.45 is

0.4975 .

The binomial standard deviation with p = 0.65 is

0.4970 , which is slightly

smaller.

Thus, data with sample proportion 55% is closer to 45% than it is to 65%.

0.45 0.55n

0.65 0.35n

n

n