Guessing the Likelihood of Events

1. Suppose an equal number of people are born on each of 365 days of each year. How many randomly selected people would have to be in a room before the probability of identical birthdays for at least two occupants was 50% ?

I think there would have to be about ?? people in the room for the probability of two with the same birthday to be 50%.

1. Suppose an equal number of people are born on each of 365 days of each year. How many randomly selected people would have to be in a room before the probability of identical birthdays for at least two occupants was 50% ?

I think there would have to be about 23 people in the room for the probability of two with the same birthday to be 50%.

2. With the same assumption, How many randomly selected people would have to be in a room before the probability of identical birthdays for at least two occupants was 99% ?

I think there would have to be about ?? people in the room for the probability of two with the same birthday to be 99%.

2. With the same assumption, How many randomly selected people would have to be in a room before the probability of identical birthdays for at least two occupants was 99% ?

I think there would have to be about 57 people in the room for the probability of two with the same birthday to be 99%.

3. The United States has 435 congressional districts approximately equal in population. Suppose they are in fact equal in population and that 435 people are selected randomly from the national population. We recognize that the average number of selections per district is exactly one, but that with a random draw some districts might have no one selected and some might have two or more. In the average case, how many districts would you expect would have no one selected?

For the average case, I think there would be about ?? districts of the 435 from which no one had been selected.

3. The United States has 435 congressional districts approximately equal in population. Suppose they are in fact equal in population and that 435 people are selected randomly from the national population. We recognize that the average number of selections per district is exactly one, but that with a random draw some districts might have no one selected and some might have two or more. In the average case, how many districts would you expect would have no one selected?

For the average case, I think there would be about 161 districts of the 435 from which no one had been selected.

4. With the same situation as in problem 4, suppose some district had six people selected. You realize that this is possible, but it may makes you doubt that the 435 people were actually selected randomly. How likely do you believe it is to have a district with six selected people? Answer with a fraction p.

If there were many randomly performed distributions, I’d expect that for ?? of the distributions, there would be a district with six people selected.

4. With the same situation as in problem 4, suppose some district had six people selected. You realize that this is possible, but it may makes you doubt that the 435 people were actually selected randomly. How likely do you believe it is to have a district with six selected people? Answer with a fraction p.

If there were many randomly performed distributions, I’d expect that for 22.4% of the distributions, there would be a district with six people selected.

V1 Bomb Strikes South of London

World War II

The data:number sites

0 229

1 211

2 93

3 35

4 7

5+ 1

R. D. Clarke: An application of the Poisson distribution, Journal of the Institute of Actuaries, v 72 (1946), p. 48.

The infected area has been divided into 576 “sites”, each of which is a ½ x ½ km. square. Then the number of bombs in each site was counted.

The data:V1 Bomb Site Frequency

0 50 100 150 200 250

0

1

2

3

4

5+

Nu

mb

er

of

Str

ike

s

Observed Number of Sites

Eight sites had four or more strikes and 229 went unhit. Were there particular targets?

V1 Bomb Site Frequency

0 50 100 150 200 250

0

1

2

3

4

5+

Nu

mb

er

of

Str

ike

s

Observed Number of Sites

A predictive model:

number sites predicted

0 229 226.74

1 211 211.39

2 93 98.54

3 35 30.62

4 7 7.14

5+ 1 1.57

What if we simply considered randomly placing 537 bombs into 576 sites?

A predictive model:V1 Bomb Site Predictions

0 50 100 150 200 250

0

1

2

3

4

5+

Nu

mb

er

of

Str

ikes

Predicted Number of Sites

Car Thefts in Avon and Somerset – 1998

An Astrological Issue?

Car Thefts

0 200 400 600 800 1000

May 21-June 21

April 21-May 20

June 22-July 23

July 24-August 23

March 21-April 20

February 20-March 20

August 24-September 23

December 23-January 20

September 24-October 22

January 21-February 19

October 23-November 22

November 23-December 22

Car Thefts per day of Period

0.00 5.00 10.00 15.00 20.00 25.00 30.00

May 21-June 21

April 21-May 20

June 22-July 23

July 24-August 23

March 21-April 20

February 20-March 20

August 24-September 23

December 23-January 20

September 24-October 22

January 21-February 19

October 23-November 22

November 23-December 22

UK Births/day 1961-75

0 500 1000 1500 2000 2500 3000

January

February

March

April

May

June

July

August

September

October

November

December

Thefts/Day and Births/Day

0.00 5.00 10.00 15.00 20.00 25.00 30.00

May 21-June 21

April 21-May 20

June 22-July 23

July 24-August 23

March 21-April 20

February 20-March 20

August 24-September 23

December 23-January 20

September 24-October 22

January 21-February 19

October 23-November 22

November 23-December 22

Are Then Birthdays Irrelevant?

Some FactsAmerican Baseball:

•More major league players are born in August than in any other month.  

•Among Americans playing major league baseball in 2005:

505 were born in August and

313 were born in July.

English Premier Soccer:

•At one point in the 1990’s, the football association’s premier league had

288 players born between Sept-Nov and

136 between born between June – Aug.

Canadian Hockey:

•Nearly 5 ½ times as many Ontario Junior Hockey League players were born in January as in November. 

•Same in National Hockey League.

What’s Going On?

The Problem with False Negative Tests

Population

No Disease

Disease

No DiseaseTest Negative

DiseaseNo Disease -Test Positive

No DiseaseTest Negative

Disease Test Positive

No Disease -Test Positive

Disease Test Negative

No DiseaseTest Negative

Disease Test Positive

No Disease -Test Positive

Disease Test Negative

Some Numbers

Suppose only 0.1% of the population has the disease.

Suppose the test’s accuracy for those who do NOT have the disease is 99.5%.

Suppose the test’s accuracy for those who DO have the disease is 99.9%.

Some Numbers

Suppose only 0.1% of the population has the disease.

Suppose the test’s accuracy for those who do NOT have the disease is 99.5%.

Suppose the test’s accuracy for those who DO have the disease is 99.9%.

Out of 1,000,000 people:• 994,005 do not have the disease and tested negative• 4,995 do not have the disease and tested positive• 999 do have the disease and tested positive• 1 does have the disease and tested negative

Some Numbers

Suppose only 0.1% of the population has the disease.

Suppose the test’s accuracy for those who do NOT have the disease is 99.5%.

Suppose the test’s accuracy for those who DO have the disease is 99.9%.

Out of 1,000,000 people:• 994,005 do not have the disease and tested negative• 4,995 do not have the disease and tested positive• 999 do have the disease and tested positive• 1 does have the disease and tested negative

Of the 5994 people who tested positive, 83.33% do NOT have the disease.

DES

DiethylstilbestrolC18H20O2

• First synthesized in early 1938 by English university research funded by the MRC (who had a policy against patenting drugs discovered using public funds).

• It was inexpensive to synthesize (from coal tar), and was produced by over 300 pharmaceutical companies.

• Its price was kept low from the beginning by competition.• DES (in tablets up to 5 mg) was approved by the FDA on September 19, 1941 for 4

indications: gonorrheal vaginitis, atrophic vaginitis, menopausal symptoms, and postpartum lactation suppression

• In 1941, DES found to be the first effective drug for treatment of metastatic prostate cancer.

• It was first prescribed by physicians to prevent miscarriages (in women who had had previous miscarriages) in the 1940s as an off-label use. On July 1, 1947, the FDA approved use for miscarriage.

• In the US, an estimated 5-10 million persons were exposed to DES during 1941-1971, including women who were prescribed DES while pregnant and the female and male children born of these pregnancies.

• In 1960, DES was found to be more effective than androgens in the treatment of advanced breast cancer in postmenopausal women. DES was the hormonal treatment of choice for advanced breast cancer in postmenopausal women for two decades.

• In the 1990s, the only approved indications for DES were treatment of advanced prostate cancer and treatment of advanced breast cancer in postmenopausal women.

• The last remaining U.S. manufacturer of DES, Eli Lilly, stopped making and marketing DES in 1997.

• Testing• DES was originally considered effective and safe for

both the pregnant woman and the developing baby. A double-blind study in 1953 of pregnant women (unselected for history of miscarriage) was not published until six years after DES received FDA approval for prevention of miscarriage. Even though it found that pregnant women given DES had just as many miscarriages and premature deliveries as the control group, DES continued to be aggressively marketed and routinely prescribed (though in decreasing frequency—sales peaked in 1953 and by the late 1960s six of seven leading textbooks of obstetrics said DES was ineffective at preventing miscarriage).

Effects:First generation• Women prescribed DES while pregnant are at a modestly

increased risk for breast cancer.

Effects:First generation• Women prescribed DES while pregnant are at a modestly

increased risk for breast cancer.• Second generation• A new study shows DES daughters as having a 2.5 fold

increase in breast cancer after age 40.

Effects:First generation• Women prescribed DES while pregnant are at a modestly

increased risk for breast cancer.• Second generation• A new study shows DES daughters as having a 2.5 fold

increase in breast cancer after age 40.• Women exposed to DES before birth (in the womb), known as

DES Daughters, are at an increased risk for clear cell adenocarcinoma (CCA) of the vagina and cervix, reproductive tract structural differences, pregnancy complications, infertility, and auto-immune disorders. Although DES Daughters appear to be at highest risk for clear cell cancer in their teens and early 20s, cases have been reported in DES Daughters in their 30s and 40s.

Effects:First generation• Women prescribed DES while pregnant are at a modestly

increased risk for breast cancer.• Second generation• A new study shows DES daughters as having a 2.5 fold

increase in breast cancer after age 40.• Women exposed to DES before birth (in the womb), known as

DES Daughters, are at an increased risk for clear cell adenocarcinoma (CCA) of the vagina and cervix, reproductive tract structural differences, pregnancy complications, infertility, and auto-immune disorders. Although DES Daughters appear to be at highest risk for clear cell cancer in their teens and early 20s, cases have been reported in DES Daughters in their 30s and 40s.

• Men exposed to DES before birth (in the womb), known as DES Sons, are at an increased risk for non-cancerous epididymal cysts and auto-immune disorders. Diethylstilbestrol can also cause feminisation of the male foetus, as DES undergoes metabolic epoxidation, and the epoxide product has affinity towards the estrogen receptors.