John Bensin's answer covers the math, but I like the plain-English examples of the theory from William Bernstein's fine book, The Intelligent Asset Allocator. At the author's web site, you can find the complete chapter 1 and chapter 2, though not chapter 3, which is the one with the "multiple coin toss" portfolio example I want to highlight.
I'll summarize Bernstein's multiple coin toss example here with some excerpts from the book. (Another top user, @JoeTaxpayer, has also written about the coin flip on his blog, also mentioning Bernstein's book.)
Bernstein begins Chapter 1 by describing an offer from a fictitious "Uncle Fred":
Imagine that you work for your rich but eccentric Uncle Fred. [...] he
decides to let you in on the company pension plan. [...] you must pick
ahead of time one of two investment choices for the duration of your
Certificates of deposit with a 3% annualized rate of return, or,
A most peculiar option: At the end of each year Uncle Fred flips a coin. Heads you receive a 30% investment return for that year, tails a
minus 10% (loss) for the year. This will be hereafter referred to as
"Uncle Fred’s coin toss," or simply, the "coin toss."
In effect, choosing option 2 results in a higher expected return than option 1, but it is certainly riskier, having a high standard deviation and being especially prone to a series of bad tosses. Chapters 1 and 2 continue to expand on the idea of risk, and take a look at various assets/markets over time. Chapter 3 then begins by introducing the multiple coin toss example:
Time passes. You have spent several more years in the employ of your
Uncle Fred, and have truly grown to dread the annual coin-toss
sessions. [...] He makes you another offer. At the end of each year,
he will divide your pension account into two equal parts and conduct a
separate coin toss for each half [...] there are four possible
Outcome First coin toss Second coin toss Total return
1 Heads Heads +30%
2 Heads Tails +10%
3 Tails Heads +10%
4 Tails Tails -10%
Being handy with numbers, you calculate that your annualized return
for this two-coin-toss sequence is 9.08%, which is nearly a full
percentage point higher than your previous expected return of 8.17%
with only one coin toss. Even more amazingly, you realize that your
risk has been reduced — with the addition of two returns at the
mean of 10%, your calculated standard deviation is now only 14.14%, as
opposed to 20% for the single coin toss. [...]
Dividing your portfolio between assets with uncorrelated results increases return while decreasing risk.
If the second coin toss were perfectly inversely correlated
with the first and always gave the opposite result [hence, outcomes
1 and 4 above never occurring], then our return would always be 10%.
In this case, we would have a 10% annualized long-term return with zero risk!
I hope that summarizes the example well. Of course, in the real world, one of the tricks to building a good portfolio is finding assets that aren't well-correlated, and if you're interested in more on the subject I suggest you check out his books (including The Four Pillars of Investing) and read more about Modern Portfolio Theory (MPT).