By John K. White
We’ve all come across either-or decisions in our lives, like the proverbial fork in the road: left-right, head-tails, red-black. They’re all two-pronged outcomes or in statistical parlance binomial events. But how do we work out the likelihood of a series of such events, say two heads in three coin tosses, three reds in four roulette wheel spins, five straight stock rises, or any of a host of new social networking interactions and computational simulations, without getting bogged down in the math? Can we make the math more visual and easier to understand?
For small multiples, we can construct a sample set, e.g., for two heads in three coin flips, where there are eight possible outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT), three of which have two heads and thus a 3/8 chance. A picture (or frequency distribution) of the possible outcomes shows the odds.
Two heads and one tail (2H, 1T) in three coin tosses (frequency distribution and Pascal’s triangle)
With larger multiples, however, this simple calculation method can become unwieldy. Another pictorial method, Pascal’s triangle, helps, named for the French mathematician Blaise Pascal who was asked to work out payouts for a friend after a gambling night unexpectedly ended, and from which the famous normal curve was later devised to work out higher multiples. But of course one still has to do the math, which although a bit hairy does the job.*
Random walks help to visualize the statistics. For example, a random walker takes 100 steps left or right. What are the odds that the walker is back at the start, that is, in the middle of the distribution? This is the same as asking the odds of fifty heads in one hundred flips or fifty stock decreases in one hundred ticks. With such pictures, we can see that the walker will most likely be back at home, but not always.
100-step 1D random walks
The same kind of analysis helped to determine that would-be French soldiers lied about their heights to escape Napoleon’s army since there were more short men than expected. Similar analysis revealed a point-spread shaving scam in U.S. college basketball games, all because the variation wasn’t varied enough. The stock market collapse of 2008 was also tied to repeated losses in an over-leveraged derivatives market, a supposed statistical blip that was meant to happen only once in a blue moon.
A 2D random walk illustrates many real-world scenes, such as clustering, urban growth, animal foraging, and even weather simulations.
1000-step 2D random walks
But can we simplify the stats even more to help understand the likelihood of multiple binomial events? Here, we turn to abstract art, using two colours randomly selected in a 10 x 10 square grid.
Random 2-colour 10 x 10 grids
As we can see, patterns appear, yet there are no 10-0 rows or columns, which we would expect about once every fifty pictures. Count how many 5-5 splits across any row or down any column, the expected maximum or mode. Or count the 4-6, 6-4, or any m-n (m + n = 10) splits. In effect, you are counting the sample set, working out the binomial odds, seeing the stats.
We must ever guard against mistaking unlikely events with probable events. Seeing the statistics helps.
White is an adjunct lecturer in the School of Physics, University College Dublin and author of Do The Math!: On Growth, Greed, and Strategic Thinking (Sage, 2013).
Further examples are available on the Do The Math! download site.
*Binomial and normal distributions:
(p+q)^n=∑_(k=0)^n▒〖(n¦k) p^k q^(n-k) 〗
y=1/(σ√2π) e^(-〖(x-x ̅)〗^2/(2σ^2 ))