|Ronny Kohavi gave a talk on "intuition busters" in March 2002 at Blue Martini Software. By chance, I later found out that most of the puzzles actually came from an excellent book by John Paulos called "Ruminations of a Numbers Man"
In this first pass, I will not supply the answers. None of these problems require knowledge of math beyond high school (except one which is currently an unsolved mystery).
|Interesting Statistical Facts:|
|Banzhaf Power Index (From A Mathematician Reads the Newspaper by John Allen Paulos) Three stockholders own 47%, 44%, and 9% of a stock. A majority of 51% is needed to pass a measure. Even though one drives a Yugo, they have equal power. Now think of four holding: 27%, 26%, 25%, 22%. The last stockholder's vote is never crucial (dummy). Similarly, 45%, 44%, 7%, 4%. The first three have equal power, the 4th is a dummy. Banzhaf power index: # of ways to change a coalition. Using this can you figure out how 3 members of the supreme court could control the vote for all 9?
Body Temperature: The normal body temperature is believed to be 98.6 degrees Farenheit by most people in the US. However, it is 98.2 degrees. Why is this number wrong? Dr. Wunderlich measurements were in Celsius, and he rounded 36.8 to 37. People in the US liked to be more accurate, so they converted 37 Celsius to Farenheit, keeping an extra digit.
Cigarette smoking kills 400,000 Americans annually. This is equivalent to three fully loaded jumbo jets crashing each and every day of the year.
Observations on Mean: Often most values are actually lower than the mean. When President Bush goes from D.C. to Texas, the average IQ rises in both states (assuming his IQ is between the mean for each state). Median home sale prices are reported in the news, so when Larry Ellison moves, the value does not spike.
To compute statistics, you sample from some universe. It is hard to get a valid sample.
Sampling: At a class reunion, the average salary is computed to claim that school X has high salary.
People who come to the reunion can afford to fly for such events.
People who don't fill the questionnaire are not making enough to brag about it.
Survey houses and you bias stay-at-homes.
Survey at night and you miss movie-goers from How to Lie with Statistics - by Darrell Huff, 1954:
Lying with Statistics:
- US Steel showed that average weekly salary increased 107% between 1940 and 1948.
What was the most important factor helping the salary increase?
In 1940 they had many more half-time employees. If you work full-time, your salary doubles.
- The death rate in the Navy during the Spanish-American War (1898) was 9 per 1000
For Civilians in New York City during the same period it was 16 per thousand.
Groups are not comparable.
The Navy is made up mostly of young men in known good health
The civilian population includes infants, old, and ill people.
- In one study, 33% of women at Hopkins had married faculty members! There were three women enrolled at the time, and one of them had married a faculty man.
- Business improved in April. Retail sales were much greater than April a year earlier.
Easter had come in March in the earlier year and in April in the later year.
|Linear versus volume: A single tower of nickels stretching from sea level to the height of Mount Everest would contain more than 4 million coins.
How large a box would you need to put these in? If you place all living humans end to end, they would extend to the moon and back more than eight times. If you tried to fit every human being into the Grand Canyon, giving each a cube, which is n x n x n, what is n?
Investing: Suppose you invest $100,000 in a volatile stock. Each year, with equal probability, it either rises 60% or falls by 40%. You declare that your heirs are not to sell the stock for 100 years. What would be the expected value of the stock after 100 years (the mean)? What would be the expected median value? What would be the expected mode?
Assume 100 people invested $100,000 in 100 different volatile stocks. The mean would be the average value of these 100 stocks after 100 years, the median would be the value in the middle if you sorted the values, and the mode would be the most common value of the 100.
Fishing pole dilemma: (I heard this one on car talk) A man bought a 5-foot-long fishing rod, then got on a bus home. The driver told him no one was allowed to bring anything longer than 4 feet on the bus. So the man went back to the store, then caught another bus, still carrying the same rod. This time, the driver said nothing. Why?
Fast Driving: A driver finished half the racing track loop at an average speed of 150 MPH
At what speed does he need to drive the other half to average 300 MPH?
Birthday Paradox: ( I know you've heard this one before, but try to remember how to calculate it - 2 ways) The probability that a random person has a birthday on 10/16 is 1/365 = 0.27%. How many people do you need to have in a room so that the probability is over 50% that two share a birthday?
Monte Hall Suppose you're on a game show, and you're given a choice of three doors:
Behind one door is a new car. Behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He then says to you, ``Do you want to pick door number 2?'
Is it to your advantage to switch your choice? Why?
(Marylin Savant showed this in parade and lots of mathematicians wrote in saying she was dumb - she was right of course) (You may remember the answer, but can you prove why)
Proof of why 1=2:
X = Y Start with this assumption
X2 = XY Multiply each side by x
X2 - Y2 = XY - Y2 Subtract Y2 from each side
(X-Y)(X+Y) = Y(X-Y) Factor
(X-Y)(X+Y) = Y(X-Y) Cancel out (X-Y)
(X+Y) = Y After cancelling
Y+Y = Y Replace X with Y (step 1)
2Y = Y Add the two Y's
2 = 1 Cancel the Y's
- now where was the error made?
2 Envelopes Paradox: You are shown two envelopes. You are told that one has X dollars and the other 2X dolars
You select one of them. The one who offered then gives you the option to exchange it for the other.
Here is why you should switch:
The one you have has Y dollars in it
There is 50% that the other has 2Y and 50% that it has 0.5Y
On average, it therefore has 0.5*2Y + 0.5*0.5Y = 1.25Y, so it has more money! Right? Why not? (this one is hard)
Simpson's Paradox: The acceptance rate for females at a small university is 56% and for males it's 70%.
Lawyers are claiming discrimination.
The university shows that in both schools (business and law), the female acceptance rate is higher
Business School: 90% for females vs. 80% for males
Law school: 33% for females vs. 10% for males
How can that be?
Potatoes: Imagine buying 100 pounds of potatoes.
You are told they are 99% water.
After leaving them outdoors for 2 days, you are told they are now 98% water.
How much do the dehydrated potatoes weigh?
Testing for Disease : Suppose you?ve taken a test for a dreaded disease D, which is 99% accurate.
If you have D, the test will be positive 99% of the time.
If you don't have D, the test will be negative 99% of the time.
Suppose that D is rare and that 0.1% of the population has D.
How concerned should you be if you test positive?