When Intuition Differs from Relative Frequency

When Intuition Differs from Relative Frequency

1. Do you think it likely that anyone will ever win a state lottery twice in a lifetime?

2. How many people do you think would need to be in a group in order to be at least 50% certain that two of them will have the same birthday?

3. You test positive for rare disease, your original chances of having disease are 1 in 1000. The test has a 10% false positive rate and a 10% false negative rate => whether you have

disease or not, test is 90% likely to give a correct answer. Given you tested positive, what do you think is the probability that you actually have disease?

Higher or lower than 50%?

“But what is chance ? What is genius ?

The words chance and genius mean nothing actually existing, and so cannot be denned. These words merely denote a certain stage in the comprehension of phenomena. I do not know how some phenomenon is brought about; I believe that I cannot know ; consequently I do not want to know and talk of chance. I see a force producing an effect out of proportion with the average effect of human powers ; I do not understand how this is brought about, and I talk about genius.”

- Tolstoy, War and Peace

Thought Questions


3. You test positive for rare disease, your original chances of having disease are 1 in 1000.

The test has a 10% false positive rate and a 10% false negative rate => whether you have disease or not, test is 90% likely to give a correct answer.

Given you tested positive, what do you think is the probability that you actually have disease? Higher or lower than 50%?

4. If you were to flip a fair coin six times, which sequence do you think would be most likely: HHHHHH or HHTHTH or HHHTTT?

Thought Questions


Are Coincidences Improbable?A coincidence is a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.

Months after $1M lottery win, US man wins $2M, Sept 28th, 2010, Associated PressJohn Wells of the Missouri Lottery said the chances of winning $1 million in the "$100 Million Dollar Blockbuster" game are 1 in 2.28 million. The odds of winning $2 million in "Mega MONOPOLY" are about the same.

The chances of winning both? Because they're independent games, it is impossible to calculate the odds, Wells said.

Couple Wins Multimillion Dollar Jackpot Twice In Ten Years, Sept 2007Adeline and Eugene Angelo won $5 million Thursday after buying the winning ticket in last week's New York Lotto. In 1996, they won $2.5 million after splitting a $10 million jackpot with three other people.

Acting Lottery Director Gardner Gurney said this was only the third time in Lotto's history that a winner has hit the jackpot twice. The odds of winning just once were 22.5 million to one.

When Intuition Differs from Relative FrequencyIntel Loses a Round in 386 Chip Battle Technology:But the fight is far from over – LA Times, Nov 1990 •Intel Corp. lost a round Friday in its dispute with archrival Advanced Micro Devices Corp. over rights to build a sophisticated microprocessor.

•The two big Silicon Valley chip makers have battled in several courts over licensing of the technology for Intel's 80386 microprocessor, the brain inside many advanced personal computers…….

•The case is a narrow offshoot of a years-long battle that grew out of a 1982 pact between the two companies, making AMD a second source for Intel chips. The agreement broke down in 1984 and the war began……

•Until last month, AMD's plans to market a clone were kept under fairly tight wraps. But a bizarre series of events led to the latest legal skirmish.

•Intel filed the suit in early October after AMD documents sent by express delivery to the Sunnyvale Hilton, and intended for an AMD employee named Mike Webb, were misrouted to an Intel employee named Mike Webb. He had earlier stayed at the hotel.

•Intel's Webb passed the documents to his boss, who turned them over to the Intel legal department. On the cover sheet, lawyers discovered references to a product called Am386.

•They resealed the package and sued AMD, citing the cover sheet as proof that AMD planned to use the 386 designation, which Intel claims as a trademark……


We often ask the wrong question …• The 1 in 17 trillion is the probability that a specific individual who plays the NJ state lottery exactly twice will win both times.

• Millions of people play lottery every day, so not surprising that someone, somewhere, someday would win twice.

Stephen Samuels and George McCabe calculated … at least a 1 in 30 chance of a double winner in a 4-month period and better than even odds that there would be a double winner in a 7-year period somewhere in the U.S.

Most Coincidences Only Seem Improbable

Coincidences seem improbable only if we ask the probability of that specific event occurring at

that time to us.

Someone, Somewhere, Someday

What is not improbable is that someone, somewhere, someday will experience those events or something similar.

When Intuition Differs from Relative FrequencyWhat looks more random?Toss a coin 4 times and record the results of each toss. Which of these outcomes are more probable?

HTHT TTHH HHHT HHHH

They are 16 possible outcomesHHHH THHHHHHT THHTHHTH THTHHHTT THTTHTHH TTHHHTHT TTHTHTTH TTTHHTTT TTTT

The probability of getting all heads is 1/16 or (0.5) (0.5) (0.5) (0.5) equal to 0.0625. The probability of getting 50% heads and 50% tails is 6/16 (0.375).

Probability Distribution for the number of headsNo. of Heads

0 1 2 3 4

Proportion:

0.0625 0.25 0.375 .25 0.0625


Now say we tossed a coin 10 times and record the results of each toss.

HHHHHHHHHH THHHHHHHHHTTHHHHHHHH….THTHTHTHTH….TTTTTTTTTHHTTTTTTTTTTHTTTTTTTTTTTT - There are a 1024 possible outcomes (2 to the power of 10)

The probability of all heads is 1/1024 or 0.001. The probability of getting 50% heads and 50% tails is 252/1024 or 0.246.

Probability Distribution for number of heads

No. Heads

0 1 2 3 4 5 6 7 8 9 10

Prob 0.001 0.010 0.044 0.117 .205 .246 .205 0.117 0.044 0.010 0.001


tossing a coin 4 times

tossing a coin 10 times




Breakdown of Actual Status versus Test Status for a Rare Disease

Test shows positive Test shows negative Total

Actually sick 90 10 100

Actually healthy 9,990 89,910 99,900

Total 10,080 89,920 100,000

Notice that of the 10,080 who test positive, only 90 are diseased. So the probability of being diseased, given that someone tests positive, is only 90/10,080 = 0.009 or 0.9%.

If base rate for disease is very low and test for disease is less than perfect, there will be a relatively high probability that a positive test result is a false positive.

You test positive for rare disease, your original chances of having disease are 1 in 1000. The test has a 10% false positive rate and a 10% false negative rate => whether you have disease or not, test is 90% likely to give a correct answer.


Breakdown of Actual Status versus Test Status for Breast Cancer

Test shows positive Test shows negative Total

Actually sick 1,260 140 1400

Actually healthy 4,930 93,670 98,600

Total 6,190 93,810 100,000

Mammogram Test for Breast CancerThe probability of woman getting breast in her 40’s is 0.014 - (from cancer.gov)

The mammogram test has a 5% false positive rate and a 10% false negative rate- conservative estimates

Notice that of the 6,190 who test positive, only 1,260 have breast cancer.

So the probability of having breast cancer, given that someone tests positive, is

1,260/6,190= 0.203 or 20%

To determine probability of a positive test result being accurate, you need:1. Base rate or probability that you are likely to have disease, without any knowledge of

your test results.2. Sensitivity of the test – the proportion of people who correctly test positive when they

actually have the disease3. Specificity of the test – the proportion of people who correctly test negative when they

don’t have the disease


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When Intuition Differs from Relative Frequency