Craig Anderson, University of Glasgow

James Howells is considering buying a council dump in south Wales after his former partner accidentally threw away a hard drive containing his bitcoin wallet. Howells has already lost a high court case to allow him to search the tip for the hard drive, which he believes contains bitcoin worth £600 million.

But would it even be possible to find it? Let’s do the maths.

Howells, a Welsh IT engineer, was an early adopter of the cryptocurrency bitcoin in December 2008. By February 2009, he had started mining the coins on his laptop – a process which involves using your computer to carry out complex mathematical processes in exchange for the coins.

At the time, he was one of just five people mining the currency, and he eventually accrued a fortune of around 8,000 bitcoin. Initially, these were basically worthless – the first real-world transaction involving the currency was in 2010, when a man in Florida bought two pizzas for 10,000 bitcoins.

However, in the 15 years since, the value of the currency has grown dramatically, with a single bitcoin passing the US$100,000 mark in December 2024 – a value which would mean those two pizzas are now worth US$1 billion (£790 million).

Doing the calculations

No wonder Howells wants to find his hard drive. But what are the chances of finding a tiny 10cm hard drive in a site containing 1.4 billion kg of waste? Is it literally like finding a needle in a haystack?

At first, this seems like a simple calculation. If we randomly select a single location within the landfill, the probability that the hard drive will be there is simply the size of the object divided by the total size of the landfill.

A Google maps estimate of the area of the Docksway landfill site suggests it is roughly 500,000 square metres (or 5 billion square centimetres), which is approximately the size of 70 football pitches.

However, we also have to account for the depth of the landfill, with years of rubbish piled on top of each other. Even a conservative estimate of 20 metres would give a total volume of 10 million cubic metres (or 10 trillion cubic centimetres). This is roughly 3,600 times the volume of the swimming pool used at last summer’s Paris Olympic Games.

Howells says the bitcoin are on a 2.5-inch hard drive, which has a volume of around 70 cubic centimetres (7cm x 10cm x 1cm). Therefore, the odds of finding the bitcoin at a single randomly selected location are 70/10,000,000,000,000 = 0.000000000007 – approximately a one in 143 billion chance.

This is over 3,000 times less likely than winning the jackpot on the UK’s National Lottery. However, with £600 million on the line, it seems unlikely anyone would just turn up and search one single location.

So, the real question here is about time and money. If we know that the hard drive is located somewhere within the landfill site, how long would it take to find it, and how much would it cost?

If we focus on time to begin with, this is really just an extension of our first calculation. Suppose it takes 1 second to search each 1,000 cubic centimetre section of the landfill (an incomplete estimate since my experience of hunting landfill for hard drives is limited), then it would take us 10 billion seconds (or 316 years) of continuous searching to cover the entire site. But of course, this could be significantly reduced by having an entire team searching at the same time.

Is it financially worth it?

Clearly, Howells does not have 316 years available to complete his search, but what if he was given the resources for one full year of non-stop searching? The odds of finding the hard drive in this year would be 1 in 316, and while the chances remain slim, this might start to sound tempting given the potential reward.

That is where the aspect of cost comes in. How much would you be willing to pay in order to have a 1 in 316 chance of winning £600m? The answer lies in the statistical concept of “expected value”“, which is the expected long-term outcome of a scenario if you were able to repeat it over and over again.

For example, suppose you were rolling a die, and you were told that you would be given £2 if you rolled a six but would have to pay £1 if you rolled any other value. You can work out the expected value of this game to see if it is worth playing. The odds of rolling a 6 are 1/6, and the odds of rolling any other value are 5/6. We can therefore compute the expected value as:

E [winnings] = 1/6 * £2 + 5/6 * (-£1) = 2/6 – 5/6 = -3/6 = -£1/2

In other words, you would expect to lose half of £1 (or 50p), on average, every time you played this game.

In the case of our bitcoins, we can think about the expected value as being the amount of money you would expect to make on average if you searched the landfill for a whole year. We would expect that, on average, we would find the hard drive (and the £600 million) 1 time out of 316, and would fail to find it 315 times out of 316 and get absolutely nothing. Therefore, we can compute the expected value as:

E [£ found] = 1/316 * £600m + 315/316 * 0 = £1,898,734

This means that on average, by searching the site for a year, you would expect to find £1.9 million. So, if the searching costs were less than this amount, you would expect to make a profit on average, and it may be considered a worthwhile investment. However, if the search cost more than £1.9 million, you would expect to lose money on average, and it would not be considered worthwhile.

These calculations can be easily adjusted to account for different lengths of search time, number of people searching, or indeed different sizes of landfill site or search area.

If Howell ever gets access to the dump, it might be worth having a statistician on hand to help guide the search (and of course, I would be happy to offer my services for a small fee…).

Craig Anderson, Senior Lecturer in Statistics, University of Glasgow

This article is republished from The Conversation under a Creative Commons license. Read the original article.