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In the debate on whether to use actively managed mutual funds or passive index funds, one argument gets scarce mention — but may be the single best reason for choosing indexing over active management.

I’d like to add one more benefit to his comprehensive list, call it the More is Less Dilemma:

The probability that a portfolio of actively managed mutual funds will outperform a similar portfolio of passively managed index funds decreases as the number of funds in the portfolio increase.

Most of the analysis in this ongoing discussion focuses on the likelihood that a single fund can outperform its benchmark index over a given time period. Proponents of active management like to focus on the short term — the time frame that is most likely to produce benchmark beating returns. Proponents of index funds like to focus on longer time horizons — the time frame that is relevant to retirement investors.

Few investors are foolish enough to place all of their retirement assets in just one fund and hope for the best. Many understand the benefits of diversification but don’t know how to diversify their assets, so they seek advice from a financial adviser.

Let’s assume your adviser recommends that you allocate your portfolio among five different actively managed funds — a domestic large-cap stock fund, a domestic small-cap stock fund, an international stock fund, an emerging-market stock fund and a domestic bond fund.

Your brother-in law, who has read all John Bogle ’s books about index investing, is content to receive market equaling returns and creates a portfolio similar to yours but uses only index funds. If only one or two of the funds in your portfolio outperforms its benchmark index, your portfolio will likely underperform your brother-in-law’s portfolio.

This would be an unacceptable state of affairs. In order to have Thanksgiving dinner bragging rights in the years to come, you want at least three of your five funds to outperform their benchmark index.

A recently issued Vanguard white paper reported that for the 15 years ending 2012; only 18% of all domestic stock funds outperformed their benchmark index and those that managed to outperform did so inconsistently. According to Standard & Poor’s. only 25% of large-cap domestic stock mutual funds, 10% of midcap funds and 17% of small-cap funds managed to outperform their benchmark index for the five years ending 2012.

So, for our discussion, I’ll assume that over the next 10 years, one in five funds (20%) will outperform a comparable index fund. Unfortunately, there is no evidence of consistent fund performance from one time period to the next, making tomorrow’s outperformers unidentifiable today.

So the question you need answered is: “If there is a 20% chance that any one fund will outperform its benchmark index, what is the probability that three or more of my five funds can do so? It seems intuitive that the chances are less than one in five but how much less?”

At this point the typical adviser, who has probably never been asked this question, will likely reply that his recommendations will be among the outperformers because he, his firm, his research team, etc. only recommend the best funds.

If you fall for this line of baloney, I feel sorry for you.

Fortunately,mathematicians found a way to calculate the answer — the binomial probability formula. Left brained number crunchers can ponder the formula here. Based on this formula, there is only a 6% chance that at least three of the five funds will outperform.

If you own more than five funds in your portfolio, what happens to your chances of outperformance?

If you own eight funds, there is a 6% chance that four or more funds will outperform their benchmark and a 1% chance that five or more will do so. And you wonder why your broker hasn’t made you rich.

The binomial probability formula reveals a Catch 22 for active investors — by prudently adding more funds to diversify their portfolio, they increase the likelihood that it will underperform a comparable indexed portfolio.

It’s foolish to play a game in which the odds of winning are so slim. Yet many investors continue to play because they, or those they trust for financial guidance, don’t understand the improbability of victory. The good news is that passively managed funds have been gaining assets at the expense of actively managed funds in recent years.

Yet the great majority of investors’ retirement funds are still actively managed. Chalk up one more victory for marketing over mathematics. The active versus passive debate should have ended long ago.

The volume of data showing the futility of active management is indisputable — to anyone who’s paying attention and whose income doesn’t depend on convincing people otherwise.

John Bogle on the rise of index funds