Sunday, February 24, 2008
As all scrabble players know, if you use all 7 of your letters, you get a bonus. However, except for the first turn, you'd need an 8-letter word to achieve this--ETAERIO would not do. Thus, I am going to try to answer the question: "What are the chances of getting the letters ETAERIO on your initial turn in scrabble (you also have to hope you are first)?" I have no hope of finding out whether this is the most likely seven letter word, because I can't automatically check all letter combinations, but I will try to give some guidance there as well.
To find the chances of gettting ETAERIO, we need the number of combinations that produce these letters divided by the total number of combinations. In other words, we have to go back to 12-th grade math, where we all learned (or sort-of learned) permutations and combinations.
There are 100 tiles in scrabble and we are choosing 7. Thus, there are 100 ways to to choose the first tile, 99 ways to choose the second, and so forth down to 94. If we chose them in order, we'd have 100*99*98*97*96*95*94 permutations. However, we don't care about the order, so we have to take the above product and divide by the number of ways we can permute the 7 tiles, which is 7*6*5*4*3*2*1. The shorthand way to express this number of combinations is "100 choose 7" or
By dividing the two ratios above ((100*99*98*97*96*95*94) / (7*6*5*4*3*2*1)), we come up with 16,007,560,800. Since most letters appear multiple times, the number of possible letter combinations is far less, and to know the chances of getting ETAERIO, I need to know how many times each letter appears.
Thus, I found the letter distributions on Wikipedia (counting our own scrabble pieces would probably not do with a three-year old distributing them around the house). The most common ones as follows:
E - 12 tiles
A, I - 9 tiles
O - 8 tiles
N, R, T - 6 tiles
D, L, S, and U - 4 tiles
other letters - 3 or less, but not relevant here
To figure out the chances of getting ETAERIO, we need to know the number of combinations that produce it. We need 2 E's, 1 T, 1 A, 1 R, 1 I, and 1 O. It turns out that the number of ways is the product of each of these implied combinations. Thus, it is "12 choose 2" (E's) times "6 choose 1" (T) times "9 choose 1" (A) and so forth. This comes out to 1,539,648 ways to get the letters in ETAERIO. If we divide this by the total number of combinations (16,007,560,800), we find that there is about a 1 in 10,000 chance of getting ETAERIO as your first 7 letters. Of course, from there, you have to know it is a word and figure out that you can make that word from those letters, since they are not likely to appear in that order.
I could not find another word with a higher probability, but I did find TREASON and TRAINED (both about 1 in 20,000). However, It's clear from the distribution of letter tiles that in order to find a word that beats ETAERIO, you can only use letters appearing in 6 or more tiles.
Now that we all remember the mechanics of combinations (or at least, we are on the subject of them), let's investigate another oft-asked question around here: what's the chance of being dealt a 7 card suit in bridge? This would be 4 (number of suits) times "13 choose 7" (ways to choose 7 from a suit) times "39 choose 6" (ways to choose the other 6 from the other 3 suits) divided by "52 choose 13" (ways to choose 13 cards from 52). This comes out to about 3.5%, or 3 or 4 times in every 100 hands.
For an 8-card suit, it is 1 in about 200. For a 9-card suit, it is about 1 in 2,700. Of course, my kids are always asking about the chances of being dealt a 10 card suit or even a 13-card suit:
10-card suit: 1 in 60,738
11-card suit: 1 in 2,746,693 (less than 1 in million)
12-card suit: 1 in 313,123,057 (less than 1 in 300 million)
13-card suit: 1 in 158,753,389,900 (less than 1 in 150 billion)
The chances aren't too great, but with some really poor shuffling, they've managed the 13-card suit once or twice.
Tuesday, February 12, 2008
Housing as an Investment
Forgetting for the moment about the psychological advantages and disadvantages of buying versus renting, let's look at the Economics, and, of course, the probabilities.
One of the most fascinating pieces of information to answer this question is a chart put together by Robert Shiller (Yale Economist) as part of his book Irrational Exuberance (he also has an article on the housing market with similar charts in Economists' Voice, March, 2006). Shiller looks at inflation-adjusted housing prices from 1890 to the present. I got the chart from this site, and the whole article is free and downloadable here (search on Shiller).
Shiller sets the price of a house in 1890 at 100, and shows how the value varies over time, adjusting for inflation. Thus, in 1947, soon after the war, the value is 110, 10% higher than in 1890. In 1989, the peak of the last boom, it is around 125. And now? Around 200!
Besides the obvious "irrational exuberance" of the housing market that is indicated in this graph, another interesting fact comes out: Housing goes up and down over time and in any given 20 or 30 year period, can be either a good investment or a bad one. Sure, if you timed the last couple of booms correctly, you could have made a killing, but the fact is that a house bought in 1960 was basically the same price in 1995, after accounting for inflation. Of course, the person who bought that house could have lived there for 35 years, paying only the cost of upkeep, and, presumably the mortgage.
Total Return to Renting Versus Buying
That brings us to the next topic: knowing that a house may or may not give you any real capital appreciation, is it better to buy or rent?
One of the big arguments I hear from my mother-in-law against renting is that "you are just throwing your money away." Seems like a good point. With renting, you get absolutely nothing out of it, but with buying, after 30 years, you own a house. The problem with this argument is it ignores two things: 1) the down payment, and 2) interest.
When you buy a house, you put around 20% down. That money then cannot be invested elsewhere. In addition, you pay interest on a loan, whose proceeds are invested in the house. The good thing is that you are using the proceeds from your loan to buy an asset 5 times the value of what you invested in cash. For example, if you buy a $500,000 house, you only have to pay $100,000. Thus, if you are in one of the boom times in Shiller's graph, you get 5 times what his graph shows in return on your $100,000 investment. The flip side, of course, is that in the bust times, you get 5 times the losses.
Compare this to renting. Here, you keep your $100,000, perhaps investing it in safe 5-year Treasury notes, where you can expect an inflation adjusted return of about 2.5% (see the fed site for T-note rates and the Bureau of Labor Statistics site for inflation rates--the real return is lower if you go back more than 40 years).
Now, let's look at renting or buying with specific numbers. Suppose that you have a $500,000 home in mind. For buying the house, your total costs are your mortgage and your return is the amount you get back after the sale, minus the $100,000 you paid as a down payment. For renting, your costs are your rent and your return is the cash you get in interest from your $100,000 investment.
The following table lays out 6 scenarios (click on the table to see a legible version). I've adjusted for the tax benefits of the mortgage as well as inflation.
I varied the mortgage interest rate and the (annual) house appreciation. The two numbers at the bottom are your out of pocket monthly costs after 5 and 10 years for owning the house. Presumably, this is the amount in rent you should be prepared to pay. Thus, for a house that appreciates at the rate of inflation (roughly what has happened with housing since 1890--it is the 3% appreciation in the 2nd and 5th columns), your monthly costs are $633 if you have a 6% mortgage and sell after 5 years and $1,134 if you have an 8% mortgage. You do much better if you hold the house for 10 years. Not true for the house that does not appreciate. Then, you do better if you hold it for less time.
To really answer the question well, we'd need an accurate prediction of inflation and mortgage rates. In the short term, both of these are pretty easy. The second thing you need is an idea of how long you will hold the house. If the house appreciates at the rate of inflation, then you are better off holding onto it for longer. If it does not, then you are better off selling sooner.
What is apparent is how radically the value changes depending on the assumptions about how much the house will appreciate. Put in some negative numbers and it really gets scary. If the current boom results in a 2% annual nominal depreciation over the next 5 years, then your 5-year monthly cost goes to about $2,500. At 5% a year depreciation (something that occurred over several years with NYC apartments in the early 90s), then your out of pocket is $3,500 per month if you sell after 5 years.
So, should you rent or buy?