No matter what any investor tells you, it's exceedingly difficult to beat the market over long periods of time. For any investor aiming to top the index, it would seem logical not to mimic that index. And since the index is a diversified portfolio of stocks, its polar opposite would seem to be a small, concentrated basket of investments. Indeed, among investors with long-term market-beating track records, I find that the very best make really big investment bets.

When 40% is better than 80%
Consider two hypothetical portfolios. Portfolio A contains 10 securities, each representing 10% of its holdings. Portfolio B contains 50 securities, each representing 2%.

In Portfolio A, four investments double in the course of a year, and the remaining six all decline by 10%. The annual return on this portfolio is 34%. If three stocks double and seven stocks decline by 10%, the annual return is 23%. Only 40% of the investor's picks were winners.

In Portfolio B, say that 10 investments go up by 20%, 30 rise by an average of 12%, and the remaining 10 decline by an average of 15%. Despite its 80% success rate, the annual return on this portfolio is a notch greater than 8%.

I've made some very general assumptions in my examples, but even when you tweak the numbers, investing success remains more dependent on the magnitude of the investment, not the frequency of success. Finding three to four investments that double in a year is asking a bit much even for the best investors, but the numbers still prove that a few big, successful investments are more than enough to produce very impressive returns. Of course, you also have to avoid making more than a few big mistakes.

It worked for the best then ...
It's obvious that Warren Buffett achieved his enormous success by making very few mistakes. However, Buffett's and Berkshire Hathaway's (NYSE: BRK-A) (NYSE: BRK-B) amazing performance over the past 40 or so years can also be attributed to about a dozen really successful investments. In the 1980s, Buffett invested roughly $1.3 billion, about a fifth of Berkshire's book value at the time, in Coca-Cola (NYSE: KO). His 8.6% stake is currently worth more than $12 billion. Meanwhile, his $10 million investment in the Washington Post (NYSE: WPO) now worth more than $1 billion.

It's working now ...
Mark Sellers, head of Sellers Capital, has been knocking the cover off the ball for the past five or so years, with annualized returns topping 20%. A year or so back, Sellers invested nearly 50% of his assets in Contango Oil (AMEX: MCF). At a recent value investor's conference, Sellers also stated that he had invested 25% of his company's assets in Lowe's (NYSE: LOW).

Mohnish Pabrai, another highly successful investor, also runs a concentrated fund, with holdings typically comprising more than 10% of assets.

Not for the faint of heart
Great investment opportunities are rare and precious. When they do appear, why devote extensive time and effort to understanding the business if it'll only produce a negligible return for you? When compelling investments present themselves, investors should welcome the opportunity, rare as it is, to bet big.

Further Foolishness:

Berkshire Hathaway is a recommendation in both the Inside Value and Stock Advisor newsletters, and The Motley Fool owns shares of Berkshire. Coca-Cola is also an Inside Value recommendation.

Fool contributor Sham Gad is managing partner of the Gad Partners Fund. He has a stake in Berkshire Hathaway. The Fool is 100% committed to its disclosure policy.