My recent randomness post hinged on people’s expectations of how long a run of heads or tails you can expect to see in a series of coin tosses. In the post, I suggested that people tend to underestimate the length of runs, but what does the fox maths say? The exploration of the numbers in this post draws on the excellent 1991 paper “The Longest Run of Heads” by Mark Schilling, which would be a good starting point for further reading for the mathematically inclined.. When I ran the experiment with the kids, I asked them to try to simulate 100 coin tosses, writing down a sequence of heads and tails. Their longest sequence was 5 heads, but on average, for 100 tosses, the length of the longest run (which can be either heads or tails) is 7. Not surprisingly, this figure increases for a longer sequence of coin tosses. What might be a bit more surprising is how slowly the length of longest run grows. Just to bump up the average length from 7 to 8, the number of tosses has to increase from 100 to 200. It turns out that the average length of the longest run grows approximately logarithmically with the total number of tosses. This formula gives a pretty decent approximation of the expected length:

average length of longest run in n tosses ≃ logn + 1/3

The larger the value of n, the better the approximation and once n reaches 20, the error falls below 0.1%.

Expected length of runs

Growth of the Longest Run

However, averages (or, technically, expected values) like this should be used with caution. While the average length of the longest run seen in 100 coin tosses is 7, that does not mean that the longest run will typically have length 7. The probability distribution of the length of the longest run is quite skewed, as is evident in the chart below. The most likely length for the longest run is 6, but there is always a chance of getting a much longer run (more so than very short runs, which can’t fall below 1) and this pushes up the average length of the longest run. Probability distribution for 100 flips

Distribution of the Longest Run in 100 coin tosses

What the chart also shows is that the chance of the longest run only being 1, 2 or 3 heads or tails long is negligible (less than 0.03%). Even going up to runs of up to 4 heads or tails adds less than 3% to the cumulative probability. So, the probability that the longest run has length at least 5 is a little over 97%. If you ever try the coin toss simulation experiment yourself and you see a supposed simulation which does not have a run of at least 5, it’s a good bet that it was the work of a human rather than random coin. Like the average length of the longest run, this probability distribution shifts (approximately) logarithmically as the number of coin tosses increases. With a sequence of 200 coin tosses, the average length of the longest run is 8, the most likely length for the longest run is 7 and the chances of seeing a run of at least 5 heads or tails in a row is now over 99.9%. If your experimental subjects have the patience, asking them to simulate 200 coin tosses makes for even safer ground for you to prove your randomness detection skills. Probability distribution for 200 flips

Distribution of the Longest Run in 200 coin tosses

What about even longer runs? The chart below shows how the chances of getting runs of a given minimum length increase with the length of the coin toss sequence. As we’ve already seen, the chances of seeing a run of at least 5 gets high very quickly, but you have to work harder to see longer runs. In 100 coin tosses, the probability that the longest run has length at least 8 is a little below 1/3 and is still only just over 1/2 in 200 tosses. Even in a sequence of 200 coin tosses, the chances of seeing at least 10 heads or tails in a row is only 17%.

Run probability profiles

Longest Run probabilities

Getting back to the results of the experiment I conducted with the kids, the longest run for both the real coin toss sequence and the one created by the children was 5 heads. So, none of the results here could help to distinguish them. Instead, I counted the number of “long” runs. Keeping the distribution of long runs for 100 tosses in mind, I took “long” to be any run of 4 or more heads or tails. To calculate the probability distribution for “long” runs, I used simulation, generating 100,000 separate samples of a 100 coin toss sequences. The chart below shows the results, giving an empirical estimate of the probability distribution for the number of runs of 4 or more heads or tails in a sequence of 100 coin tosses. The probability of seeing no more than two of these “long” runs is only 2%, while the probability of seeing 5 or more is 81%.

These results provide the ammunition for uncovering the kids’ deceptions. Quoting from the Randomness post:

One of the sheets had three runs of 5 in a row and two runs of 4, while the other had only one run of 5 and one run of 4.

So, one of the sheets was in the 81% bucket and one in the 2% bucket. I guessed that the former was the record of coin tosses and the second was devised by the children. That guess turned out to be correct and my reputation as an omniscient father was preserved! For now.

Runs at least 4 long

If you have made it this far, I would encourage you to do the following things (particularly the first one):

  1. Listen to Stochasticity, possibly the best episode of the excellent Radiolab podcast, which features the coin toss challenge
  2. Try the experiment on your own family or friends (looking for at least 3 runs of 5 or more heads or tails and ideally at least one of 6 or more)
  3. Share your results in the comments below.

I look forward to hearing about any results.

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I have been watching some (very) old Doctor Who episodes, including the first ever serial featuring the archetypal villains, the Daleks. In this story, the Daleks share a planet with their long-time enemies, the Thal. After a war culminating in the denotation of a neutron bomb, both races experience very different mutations. The Daleks have become shrunken beasts that get about in robotic shells, while the more fortunate Thals mutated into peace-loving blondes.

The Thals hope to make peace with the Daleks, but the Daleks have more fiendish plans and plot to lure the Thals into their city with a gift of food and then ambush them. It is a good plan, but it is the choice of gifts that left me bemused. There is plenty of fruit and some large tins whose contents remain undisclosed. These may be reasonable choices, although I do find it hard to picture the Daleks stacking melons with their plunger hands. But the trap also appears to feature stacks of toilet paper. Granted, toilet paper may be an appealing luxury for the Thal, who have been trekking through the jungle for a year, but the real question here is, why do Daleks even have toilet paper?

Dalek ambush

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Randomness

6 April 2014

With three children, I have my own laboratory at home for performing psychological experiments. Before anyone calls social services, there is an ethical committee standing by (their mother). This evening, I tried out one of my favourites: testing the perception of randomness. Here is the setup: I gave the boys two pieces of paper and […]

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Chinese non-residents…in China

31 March 2014

Recently I travelled to China for the first time. My first glimpse of Beijing took in the Escher-like headquarters of Chinese TV station CCTV. It is an extraordinary building and to get a proper sense of it, you have to see it from a number of different angles. Driving across the city, impressed by the […]

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Bringing Harmony to the Global Warming Debate

25 February 2014

For some time now, our regular contributor James Glover been promising me a post with some statistical analysis of historical global temperatures. To many the science of climate change seems inaccessible and the “debate” about climate change can appear to come down to whether you believe a very large group of scientists or a much […]

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I’m with Felix

16 February 2014

FT blogger Felix Salmon and venture capitalist Ben Horowitz have very different views of the future of Bitcoin. Salmon is a skeptic, while Horowitz is a believer. A couple of weeks ago on Planet Money they agreed to test their differences with a wager. Rather than a simple bet on the value of Bitcoin, the bet centres […]

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Where Have All The Genres Gone?

1 February 2014

The Mule has returned safely from the beaches of the South coast of New South Wales. Neither sharks nor vending machines were to be seen down there. We did, however, have a guest drop in. none other than regular blog contributor, James Glover. The seaside conversation turned to music and James has distilled his thoughts […]

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Shark season

31 December 2013

Summer in Australia comes with cicadas, sunburn and, in the media at least, sharks. So far, I have learned that aerial shark patrols are inefficient (or perhaps not) and that the Western Australian government plans to keep swimmers safe by shooting big sharks. Sharks are compelling objects of fear, right up there with spiders and snakes […]

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Power to the people

5 December 2013

Regular Mule contributor, James Glover, returns to the blog today to share his reflections on solar power. I have been investigating solar power for years and finally bit the bullet and signed up for a system. A 4.5kW system cost me $8,500, after receiving the Government rebate (about $3,000). I’ve been meaning to write about […]

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Qantas and Adobe

9 November 2013

In my last post, I complained about the approach Qantas has taken to password security for its new Qantas Cash website. When I called Qantas to express my concerns, my query was referred to the “technical team”. I was assured they would be able to assuage my concerns. Here is the email response I received: […]

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