Suppose you have a bet which loses money most of the time, but wins a massive amount now and then, how much money should you put on it?

Let's say the 1 time you win, you win $a for each dollar you bet, and the N times you lose, you lose $b for each dollar you bet. By the Kelly criterion, the geometric average rate of gain if you bet $x$ of your wealth would be

$$R = (1+xa)(1-xb)^n$$

Setting $\partial_x R = 0$, you get

$$x = \frac{a - nb}{(n+1)ab} = \frac{\left<\mathrm{arithmetic\ gain}\right>}{ab}$$

Suppose you are asked to flip a coin, and heads you win $3, and tails you lose $1---then $n=1, a=2,b=1$, and therefore $x = \frac{1}{2 \cdot 2} = \frac{1}{4}$, i.e., you should bet 25% of your wealth.

If you have a lottery ticket that has a 1 out of 5,000 chance of winning $10,000 that costs $1, and you are only allowed to buy one number, then $n=4999,a=9999, b=1$, and
$x = \frac{5000}{49999 \cdot 9999} \approx 10000^{-1}$ and you should only bet 0.01% of your wealth at a time.

Conversely, if you were selling a lottery ticket that had 1 out of 10,000 chance of winning $5,000 that cost $1, $n = 9999, a = -4999, b = -1$, and $x = \frac{5000}{9999\cdot4999} \approx 10000^{-1}$ and you should be trying to have about 0.01% of your wealth at stake.

Related: Do not play the lottery unless you are a millionaire

When I first found out about Case-Shiller futures, I was pretty excited at the possibility of buying a house, enjoying the cheap loan and tax benefits, and hedging out most of the risk.

Alas, the futures have no volume. As of today the open interest on the Feb 2010 New York Case-Shiller futures (NYMG10) is 2 (as in the first integer greater than 1).

It's difficult to make markets when the underlying and the instrument differ so much in liquidity? Granted, SP500 futures are more liquid than the basket of stocks too, but that gap is bridgeable. What differentiates a bridgeable from an unbridgeable gap? What implications does this have for the existence of noise traders?

Money is a medium of exchange. When exchanging A for B, we prefer to exchange A for money and then money  for B.

Money is a store of value. On top of holding money temporarily while exchanging A for B, we also hold money when we haven't determined what B we want yet. Holding money is preferable to holding A because A might be bothersome to store (e.g., a truckload of sand) or it might become less valuable with time (e.g., a truckload of apples).

The usefulness of money gives rise to a liquidity preference. Keynes enumerated three ways in which money is useful: as a buffer to smooth out short-term volatility (known unknowns) in income and expenditure, for use as emergency reserves (unknown unknowns), and for use in speculation, i.e., using knowledge of prices to buy assets at low prices and sell them at high prices. The first two forms of liquidity preference tend to grow with income, whilst the last is more affected by the interest rate and expectations of future interest rates.

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When the economy is in a state of equilibrium, each party holds a constant amount of money, and it is possible to think of the flow of money as consisting of many cases of multiparty barter, i.e., every dollar flows in a circle, with goods and services flowing in the opposite direction. This money flux is the GDP, and is a measure of economic activity.

Some goods are not directly consumed, and instead are used to produce other goods and services - these are called investment goods. The accumulation of investment goods increases production.

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When stimulating the economy by printing money, one dumps money into certain regions, and that money proceeds to flow outwards from the introduction points. Iff that flow results in the accumulation of investment goods, the economy is successfully stimulated.