I have an annoying (self-diagnosed, but likely supported by a jury of my peers) habit to have my interest piqued when I hear certain terms. I’m not sure I can provide an exhaustive list, but typically anything involving experimental design, statistics, or numbers will cause all other thoughts to cease in order to narrow in on the conversation. Usually what happens is I begin to ask a barrage of questions, such as:

• Can you elaborate a little more?
• How do you define x?
• Why do you think that is?
• Have you thought about x?

I always joke that my academic and professional background, along with my innate interest in problems and numbers, has molded me into a professional devil’s advocate. It is about endearing as it sounds, and I thank my family and friends for not completely abandoning me. But alas, it is part of who I am and how I found myself discussing randomness with my neighbor when he made the mistake of asking me to help him plant some Zinnias.

I will not profess to have anything resembling a green thumb. My parents have always gardened, but my role was relegated to the no-knowledge department – pick up the heavy stuff and move it, dig holes, and weed. It’s not that I don’t appreciate a good garden – I just never adopted landscaping as a hobby. My wife and neighbor on the other hand have made the front of both of our houses look amazing – and I’m still the willing helper to move all of the mulch and dig all of the holes.

So when my neighbor asked me to help him plant some Zinnias (red, purple, and yellow), I did what any novice should do – ask questions. The first question I asked, “How do you want to arrange the flowers? Do you have a pattern in mind?” My poor neighbor’s response – “No, I don’t want too many of the same colors next to each other. I want them planted randomly…”

Cue my walk out music…

Randomness

Randomness is such an interesting and important phenomenon. The concept has incredible importance for both experimental design and probabilistic statistics. When we assess the outcome of our statistical analysis, we are making direct inferences about our findings in the face of what we know to be random. In its simplest form, are our findings interesting/explainable or are they just random?

What makes random even more interesting is our difficulty in truly discerning random from non-random events.^1,2 In fact, we are drawn to patterns because we naturally want to make sense of the world. Additionally, we expect patterns to be meaningful – even when they are not. Take for example the Gambler’s fallacy (also known as the Monte Carlo fallacy). If the Roulette Ball lands on Black five times in a row, you might be tempted to say, “well it has to be Red this time – put all my money on Red.” However, these events are independent. Nothing about the previous attempts matter for your big bet. Here is another example. Imagine I tell you that I’m going to flip a coin 10 times. The first 9 flips all land on tails. Pretty impressive! For the final flip, I ask you, “what is the probability the coin will land on tails?” You may be tempted to think it is some infinitesimal number. Nine tails already – it has to be a heads! But think carefully here – I didn’t ask you what is the probability of getting 10 out of 10 tails. I’ve asked, what is the probability that the final coin flip is tails. The answer, which may be surprising, is 50%. It is still an independent trial that has an equal probability of occurring. The point here is that understanding randomness is difficult – we want to believe patterns are meaningful and we find it difficult to believe that clusters can occur by chance.

“[P]eople expect even small samples to closely represent the properties of randomly generated data, although small randomly generated samples often contain structure by chance.” ³

Randomness and Coin Flip Clusters

If I let you flip a coin 100 times and you hit a streak of 5 tails, is the coin loaded? Based solely on this information, the answer is no. The probability of a streak of at least 5 tails in 100 coin flips is 81%. Take a look at this visual example of penny flips:

Visit https://www.omnicalculator.com/statistics/coin-flip-streak for your own streak generation probability or https://www.random.org/coins to visualize random coin flip samples.

A really illuminating example for me is from Blastland and Dilnot’s (2010) The Number’s Game: A Commonsense Guide to Understanding Numbers in the News, in Politics, and in Life. In a chapter discussing cancer clusters, the authors ask readers to imagine a scenario where one throws a handful of rice into an open room (assuming wind, a level floor, etc. were not confounding factors). Finding a cluster of rice doesn’t negate the random distribution of rice.

Which brings us back to my neighbor. Remember, he wanted the flowers planted randomly but he also wanted to ensure clusters of flowers were not planted together. But that isn’t exactly how random works. Random would mean the absence of a predictable pattern. However, clusters can still exist. How would a random garden or yellow, purple, and red flowers turn out? Give the simulation below a try! Pay attention to instances of clusters.

What did you see? If you sampled a few garden patterns, you may have found some that looked decently distributed. But if you sampled your flowers a few times, you may have seen some results that were similar to one like the image below. Although the placement is random, it looks liked I had a desire to place my yellow bulbs on the right half of the garden bed. But that’s the point. These are clusters within a random selection.

Luckily, my neighbor humored me and allowed for a non-random planting strategy. With his green thumb and my propensity to over-analyze, I’m sure our garden will be a success.

As far as my gardening ability after the Zinnia planting project…

1. Bar-Hillel, M., & Wagenaar, W. A. (1991). The perception of randomness. Advances in applied mathematics, 12(4), 428-454. doi: 10.1016/0196-8858(91)90029-I
2. Hahn, U., & Warren, P. A. (2009). Perceptions of randomness: why three heads are better than four. Psychological review, 116(2), 454-461. doi: 10.1037/a0015241
3. Williams, J. J., & Griffiths, T. L. (2013). Why are people bad at detecting randomness? A statistical argument. Journal of experimental psychology: learning, memory, and cognition, 39(5), 1473-1490. doi: 10.1037/a0032397