What it looks like:
Supplies you will need:
PROCEDURE:
1) Cut the wooden strips for the side, bottom, and bins. The number of bins is up to you (I chose 12 bins, requiring 11 wooden bin-separators and saving one extra) and can depend on the size of your board and of the objects you are going to drop down it.
2) Cut the wooden dowels into many short pegs.
3) Glue the long edges onto the sides of the board.
4) Take the number of bins you have and sketch their positions on the board, arranging them so that each bin has equal size. Glue the bin-separators in place.
5) Plan where the pegs should go: There should be one peg directly above each bin separator (so that no pennies land lying flat on one of the separators, but rather falls into one or the other). The spacing between pegs is thus determined by the spacing of the bins. The next "layer" of pegs should have the same separation, but offset by half a separation. That is, the next layer should form equilateral triangles with the previous layer. Draw each peg position in the triangular pattern.
The spacing between pegs should be about twice the diameter of the objects you're going to drop down the board.
6) Drill a hole at each drawn peg position with a hand power drill. The diameter of the hole should be the same as the diameter of the peg (in my case, 3/16").
7) For each peg, dip one end lightly in wood glue and start it into the drilled hole. Hammer it in (it should fit nice and tightly).
8) A 2-foot strip of wood goes at the bottom of the board to catch the pennies. Attach hinges to the strip first (it's easier to do this when the strip is still separate) and then attach to the bottom of the board with the other side of the hinges.
9) Screw and hook-and-eye into the side of the bottom piece. Screw a small hook into the side of the board, so that the hook slides into it when the bottom piece is in the "closed" position.
10) Glue plexiglass to the front of the board, being careful that the bottom piece can still swing open and closed.
11) Attach a funnel to the board, above the topmost middle peg.
Most should land in the middle, but some will land to the sides.
If the number of pennies you drop becomes large, you should observe the
relative numbers of the pennies falling in each bin to start to resemble
a Gaussian distribution. Also called a "normal distribution"
or a "bell curve," this is a commonly-encountered distribution in nature
for random processes.
Now let's send it through N levels of pegs. At each level, the penny will be "knocked" either to the left or to the right, each with a 50/50 probability.
p(right) = 0.5 p(left) = 1-p(right) = 0.5The probability of any one given sequence of n1 lefts and n2 rights (in a particular order) is
p(left)^n1 p(right)^n2.But there will be many ways of taking n1 lefts and n2 rights over N levels.
N! / (n1! n2!)After n1 lefts and n2 rights, the horizontal position is
m = n1-n2 = 2n1 - NPutting it all together, the probability of ending up at position m to the left of the center is:
N! (1/2(N+m)) (1/2(N-m)) p(m) = ---------------------- p(left)^ p(right)^ (1/2(N+m))! (1/2(N-m))!
-(m-m0)^2
1 {-----------}
G(m) = -------------- e^ 2 sigma
sqrt(2pi)sigma
In this experiment, N is not terribly large and the bins are so large that
the resulting histogram is very discrete and not continuous, but its general
shape should still approach that of a Gaussian.
Here's a quick first-attempt histogram with my prototype board:
If you drop 100 pennies, how many pennies would you expect to land three sigma away from the mean?
The idea here is: bins in the middle are more likely to win. So a student is smart if he or she picks a middle bin. But if all the students are smart and pick the middle bin, everybody has to share the winnings and nobody makes as much money. So if everyone is picking middle bins, a few smart students will figure out that if they pick an unlikely bin and get lucky, they get to keep the winnings all to themselves. Although the chances are less, the payback if you win is higher.
As the students play, they will figure out how to bet wisely; if not many people are in the likely bins, they stand a high chance of winning by betting there, but if there are already many people in the likely bins, then betting on an unlikely bin gives a greater payback. Eventually, all the students will be spreading their bets in a Gaussian distribution themselves in order each to optimize their winning!
This system resembles the horseracing system: Odds are given for each horse, if you bet on a horse with high odds, you win more often but share your winnings with lots of other people. If you bet on a disfavored horse, you won't win often but will keep more of the winnings when you luck out.
"You have the option of spending a dollar to play this game. You pick a bin and if you guess right, you win the jackpot. How big would the jackpot have to be before you would play this game?
What if I picked the bin for you? I choose the middle bin. If you played the game on the middle bin, what is the probability that you would win? How much would the jackpot be before you would think it was worth it to spend the dollar?
What if I picked one of the outer bins? What's the probability that you would win there? How much would the jackpot have to be before you'd be willing to spend the dollar on the outer bin?"
The idea here is to equate risk with reward in gambling. If you're going to take a greater risk, you should demand a greater reward if you win. In fact, you can calculate the reward necessary for a certain risk: If the odds are 1 in 20 of winning, then if you pay one dollar to play, the reward (jackpot) should be $20. That way, if you played 20 times you would break even. If the jackpot is more, you could play repeatedly and make a profit. If the jackpot is less, your best interest is not to play at all.
In casinos, of course, the games are designed so that the reward is always below the risk (even only slightly). Thus the house always wins, and you should stay at home and keep your money.
After all, this is what real neutrino astronomy is like. Our reconstruction is never perfect... it knocks the track direction away from the true direction (sometimes a little, sometimes a lot) just like the pegs on the board. We can't know what's really out there, just what we observe in the final bins.