|Don't read this until you've at least tried the puzzles.|
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? 6 feet.
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? 20 feet
Investing: 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 stock value (mean) after 100 years? What would be the median? What would be the mode?
The expected value would be $1,378,000,000. If there are many stocks like this, the total market value will rise dramatically (value = expected * num_stocks)
100,000 * ((1.6+0.6)/2)^100 = 100,000 * 1.1^100
The mode and the median are both $13,000. = 100,000 * (1.6)^50 * (0.6)^50.
While on average you expect 10% return a year, the most likely scenario (mode) is that you'll end up with $13,000.
Moreover, more than half the people will end up with $13,000 or less.
The moral of the story is Diversify! Now you know why you hear that investment advice from the experts so often.
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?
The man got a 3' by 4' box at the store, in which he placed the rod diagonally. The second bus driver saw this as a 4-foot object.
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?
X = length of half the track
Time to finish all = 2X/V = 2X/300 = X/150
Time spent = X / V = X / 150.
He has zero time to finish the second half.
If he drives 450, he?ll average (2X/(X/150)+(X/450)) = 225 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?
23 people are needed.
With 42 people, you have 90% probability.
23 people -> 253 pairs (23*22/2). 99.726% ^ 253 = 50% or 364/365 * 363/365 * 362/365.
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 probably remember the answer, but can you prove why)
Due to symmetry, assume you pick door #1.
Consider the three equally probable scenarios. In 2 of them, you win if you change. Therefore the probability is .66 that you win if you change. It is much better than the .33 you have if you don't change. The host gives you information when he opens a door known not to have the prize.
If you are still not convinced, consider it this way:
Suppose that there are 1000 doors. You select one of them. Now the host opens 998 of the other doors, revealing 998 goats. Do you still want to keep your door? Your door has only a .001 chance of being correct while the alternative has a probability of .999
Still not convinced? Try this simulation of the Let's Make a Deal game with Monte Hall. or this version.
Of course its possible that the host is trying to trick you. Perhaps they only open the other doors if the one you picked was the correct one. Then if you know this puzzle, you would switch, and as a result lose.
Here's a simulation of casino gambling that is mostly unrelated, but very interesting.
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?
On the fifth step you are dividing by 0 if your assumption that x=y is true. This is illegal in math.
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).
The assumption violates an esoteric rule of statistics that the sample set needs to be bounded. There are many scholarly papers written on this. Its something to do with the probabilities of infinite sets. I don't know more.
Simpson's Paradox: The acceptance rate for females at a small university is 56% and for male 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?
The business school accepts almost everyone (82.5%)
The law school accepts only a few (27.5%)
Women apply more to the law school.
While their acceptance rate is higher in both schools, the overall probability is lower.
Conclusion: when you are shown a probability, you must check for hidden variables.
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?
1% of 100 pounds = 1 pound essence.
2% of w = 1 -> 1/0.02 = 50 pounds.
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?
Don't be overly concerned
You only have 9% chance of having D.
Population of 100,000
0.1% = 100 people have D
99 of the 100 with D will test positive
However, 1% of 99,900 will also test positive, so 999 healthy people
The probability of having D is therefore
99/(999+99) = 9%
For those who understand Bayes Rule : P(D|+) =
P(+|D)*P(D)/(P(+|D)*P(D)+P(+|not D)*P(not D)) =
99%*0.1%/(99%*0.1% + 1%*99.9%) = 9%