John Allen Paulos is a mathematician, but he is also human. As a mathematician, he has a pretty good theoretical grasp of the stock market. As a human, he is prone to the same sorts of irrationalities the rest of us are. And so, three years ago, around the peak of the market, he made a series of increasingly foolish moves that culminated in his financial humiliation and his decision to write this interesting book.

It all started in early 2000, when Mr. Paulos received a "small and totally unexpected chunk of money." Instead of putting it in an index fund -- his investment policy until that point and one he knew to be theoretically sound, if a bit boring -- he decided to buy some shares of WorldCom. The market was booming, and Mr. Paulos was seduced (as which of us hasn't been?) by George Gilder and his lyrical evocation of the coming "telecosm." Why not have a little fun?.

But then Mr. Paulos began to lose his reason. As WorldCom's price dropped, he "averaged down" by purchasing cheaper shares. Swayed by the happytalk of Jack Grubman, the telecom analyst, he filtered out bad news. Soon he was buying on margin. The day his first margin call came, he happened to be browsing at a bookstore, where he came across the phrase "staying in the game." Thus confirmed in his reckless resolve, he continued to buy on margin -- taking comfort that Bernie Ebbers was reportedly doing the same. Finally, as WorldCom shares headed inexorably toward nothingness, Mr. Paulos e-mailed the great Ebbers himself, offering his services as a wordsmith to help make the public realize how dreadfully undervalued the company was..

Well, Mr. Paulos lost his mad money and more, but he deserves to recoup some of it with "A Mathematician Plays the Stock Market" (Basic Books, 216 pages, $25). If the book were just an extended autobiographical anecdote -- author gets into scrape, comes through humbled but OK -- it would be of finite interest. But Mr. Paulos, who teaches mathematics at Temple, has a knack for making technical concepts clear and entertaining, as he has shown in earlier books like "Innumeracy." Here, with his usual light touch, he lays out the mathematical ideas behind the market, cutting back and forth between the lessons of theory and his own misadventures in practice..

We get everything from the old (but still infamous) "beta," a measure of a stock's volatility relative to the market as a whole, to the latest "nonlinear" models of Wall Street's oscillations, with a financial paradox or two thrown in. But the central idea that Mr. Paulos explains is the one he himself flouted: the Efficient Market Hypothesis. Also known as the Random Walk Hypothesis, this says that a stock's price at any given moment reflects all available information; hence its future movements are unpredictable, and there is no way (other than sheer luck) to beat the market..

Thought up by a French mathematician a century ago and rediscovered by American finance professors in the 1960s, the Efficient Market Hypothesis has never been popular with Wall Street pros or with amateur investors. That, as Mr. Paulos notes, is no accident. By its very logic, it is true only when most people believe it to be false. In other words, the stock market is efficient only when lots of investors, convinced that it can't be, busy themselves looking for trends, analyzing balance sheets and trading feverishly. If everyone comes to believe the hypothesis is true, contrariwise, then they'll all lazily put their money in unmanaged index funds, and the market will cease to incorporate new information, becoming inefficient..

How does our author feel about the Efficient Market Hypothesis? "I think it holds, but only approximately and only most of the time," he says. There are some investment strategies that do, contrary to the hypothesis, seem to beat the averages, if by the slightest of margins, and Mr. Paulos presents the hard evidence in their favor. Here he does a nice job. But he can also be slapdash, managing to misspell Warren Buffett's name. His rampant jocularity will not be to everyone's taste. ("The result . . . may well be egg on one's face and the transformation of one's nest egg into a scrambled egg if not a goose egg.") And, for a short book, there is a good bit of padding, mostly in the form of the author's moralizing. Do we really need to hear about Tyco CEO Dennis Kozlowski's $6,000 shower curtain again?.

But back to the serious stuff. Mr. Paulos mentions the case against the Efficient Market Hypothesis that was assembled a few years ago by Andrew Lo of MIT and Craig MacKinlay of the Wharton School. What these researchers found was that there are indeed nonrandom trends in the market -- trends that, in theory, could be exploited by investors looking for "excess profits." In the short term, overall market returns are positively correlated: That is, a sunny day is likely to be followed by another sunny day. In the long run, stock prices display a slight negative correlation: Today's winner is a bit more likely to be a loser three to five years hence..

Messrs. Lo and MacKinlay presented their findings in mid-1999, in a book titled "A Non-Random Walk Down Wall Street." Back then, I recall, an interviewer asked Mr. Lo -- a mathematician by training, like Mr. Paulos -- the obvious question: How might an investor armed with this insight beat the market? "Today," he replied, "the opportunities are greatest in the Internet sector, because it is a new business in which the traditional valuation models don't apply.".

As Homer Simpson would say: "D'oh!".

Mr. Holt is a writer in New York.