Maths on the Back of an Envelope: Clever Ways to (Roughly) Calculate Anything
Page 10
Those are very similar to the sort of odds that are quoted for winning a lottery.
But there is a big difference between holes in one and the lottery, because, as we’ve seen earlier, while the odds in a lottery are fixed and can be worked out exactly, the odds of a hole in one are more about sticking a finger in the air and making a lot of assumptions.
For a start, those odds depend on the player. A top golfer such as Rory McIlroy or Tiger Woods is far more likely to hit the pin than a regular club player.
Then there’s the length of the hole. In order to get a hole in one, you need to be capable of hitting the ball from the tee to the green. For almost all golfers this means holes in one can only happen on shorter holes where a decent player is expected to take three strokes to get the ball into the hole (these are known as ‘Par 3 holes’). These holes are typically between 100 and 200 metres long. The shorter the hole, the easier it is to get a hole in one because an error in the direction that you aim the ball is not punished so severely.
There are usually four Par 3 holes on a golf course, so in a round of 18 holes, there are four opportunities to get a hole in one. This means that the chance of getting a hole in one during a round of golf is about four times higher than the chance of achieving it on a particular, named hole.
The ‘One in 17 million’ figure came from the National Hole-in-One register. Based on statistics that they have accumulated from around the world, they reckon that the chance of getting a hole in one on a specific, named hole is around 1 in 2,500 for a professional golfer, and about 1 in 12,000 for a club golfer. So, on a given hole, we might expect the chance of two average women golfers both getting a hole in one to be:
That’s more like 1 in 150 million.
But further investigation into the story reveals that the hole in question had been shortened to just 90 yards because the course was being repaired, which must have hugely reduced the odds of a hole in one. And also, the women were playing in a group of four, whom we will call A, B, C and D. This means there were six possible pairs of women who could have got a hole in one: AB, AC, AD, BC, BD and CD. And it would have been a headline if any of those six pairs had been successful, so we can divide the odds by six. All of which brings our 1 in 150 million down to something much smaller.
In the end, nobody really cares if the odds are 1 in 17 million or 1 in 50 million: it’s merely a chance to report that something freakishly unlikely just happened. But was it really freakishly unlikely? Before I went in to record a radio item about this story,7 I thought I’d drop into my nearest golf club, in Dulwich in south London, to see if I could pick up any anecdotes. I spoke to the manager.
‘Holes in one? We get about ten of them a year. In fact, we had an eleven-year-old get one last Sunday,’ he said, and he showed me the card sitting on his desk.
‘Oh, but if this is about those two ladies who got holes in one last week, we can beat that.’ He led me to a plaque on the wall outside his office. With a caption ‘Halved with holes in one!’ it showed two smiling men who had just tied in a match-play tournament with holes in one. This had happened in 1984.
So, the first random golf club I turned up at was able to produce an equally freakish story to the one experienced by the ladies foursome in 2017.
I made a quick back-of-envelope calculation:
No. rounds played in Dulwich per year 30,000
No. Par 3 holes played per year 30,000 × 4 ~ 100,000
No. Par 3 holes in 30 years 30 × 100,000 = 3 million.
In other words, in 30 years there have been about three million opportunities for two people to get a hole in one at the same time, and it has happened at least once. It confirms that the odds of this event are probably one in a few million.
However, since there are estimated to be over 500 million rounds of golf played around the world each year, we’d expect the story of the two ladies and the holes in one to be replicated several times each year.
And sure enough, this turned out to be the case.
To find more details about the Berkshire ladies hole-in-one story, I searched online using the phrase ‘Two women golf hole in one’. The first story to pop up was not about the Berkshire ladies. It was an almost identical story from Northern Ireland that had happened a couple of months earlier. This time it was Julie McKee and Mandy Higgins who had got holes in one, again as part of a foursome. Their fluke was described as a one-in-a-million story. Which just goes to show how hand-wavy these headline odds can be.
WHAT’S THE CHANCE OF DEALING OUT FOUR ‘PERFECT’ HANDS?
We’ve seen some remarkable coincidences already, but on the face of it, one of the most remarkable coincidences of all time happened in Kineton, Warwickshire, in April 2011. Four Warwickshire pensioners were playing a game of whist, a traditional card game in which the entire pack of 52 cards is dealt out between four players. Each player receives a hand of 13 cards.
The pack was shuffled and then dealt. To their utter astonishment, when they picked up their cards, each of the four players discovered they had been dealt a complete suit.
One of the pensioners, Wenda Douthwaite – who was dealt a hand containing all 13 spades – declared herself to have been ‘gobsmacked’.
‘I’ve never seen anything like it before’, she said. And Wenda’s surprise was justified, because mathematically, the chance of this happening in a random deal of a regular, thoroughly shuffled pack of cards is a staggering one in 2,235,197,406,895,366,368,301,559,999.8 In other words, there are about two octillion different combinations of cards that can be dealt out into four equal hands, and in only one of those does each player get a complete suit.
Let’s just look at how unlikely this was to happen.
There are currently over seven billion people on the planet. Imagine giving each of them a pack of cards, and getting them to shuffle it thoroughly and deal out four hands.
Now get them to repeat that every minute, so 60 deals per hour. Let’s allow them nine hours in the day to sleep and eat. That leaves them the remaining 15 hours a day to deal cards.
They can deal out:
15 × 60 = 900 deals per person per day.
So the total number of deals in a year would be:
900 × 7 billion × 365
h 2 quadrillion (that’s 2 × 1015) deals per year.
Even if every hand dealt was different, to deal all of the two octillion possible sets of four whist hands would therefore take:
Scientists reckon that the solar system will come to an end eight billion years from now, and so we can safely say that, according to those odds, not only is it unlikely that this hand has ever been dealt before in the history of card-playing, but even if everyone on the planet were to shuffle and deal a pack of cards every five minutes, it is unlikely to ever happen again before the universe comes to an end.
Which is very curious, because in 1998, in Bucklesham, Suffolk, four pensioners playing a game of bridge experienced exactly the same coincidence. Hilda Golding, one of the players involved on that occasion, said at the time: ‘I was amazed. I’d never seen anything like it before, and I’ve been playing for about 40-odd years.’
And if you trawl through the archives, you’ll discover other reports of the same thing: in Pennsylvania in March 1938, Virginia in July 1949, Wyoming in April 1963 and many, many more. Among the reports of four players being dealt perfect hands is one from St James’ Club in London, in 1959 … on 1 April. That date rings some alarm bells.
The point is that each of these occurrences was supposedly a once-in-the history of the universe event. The numbers do not make sense: the chance of all of these coincidences happening was so small that we are justified in calling them impossible.
Since the figures do not make sense, there must be another explanation as to what was happening.
There are two possibilities.
The first is that the cards being dealt had not been completely randomly shuffled. A new pack of cards comes with the cards neatly ordered
by suit – all the spades, then all the hearts, the clubs and the diamonds. If you cut the pack exactly in half, and do two perfect riffle shuffles, so that the two halves of the pack perfectly interweave with each other, then the pack will now be arranged in the order: spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds and so on, all the way through the pack. If you now deal the cards out to four people, the first player will get all the spades, the second will get all the clubs, and so on. I’m not saying this is what happened. But it might have done.
Even when the cards have been shuffled, when they are gathered together at the end of a hand, they will tend to be grouped together by suit.
It takes at least seven good (but not perfect) riffle shuffles to be reasonably confident that the cards have been fully mixed up, and even then, they may well retain some trace of a pattern (spades clumped together, for example).
So in all of the stories of the freak card deals, it is extremely likely that after shuffling, the cards being dealt were not in a ‘random’ order at all. And while a perfect deal would still be extremely rare, the chances of it happening when the cards have a trace of order in them is astronomically higher than if they are completely mixed.
And there is a second possible explanation. How hard would it be for a practical joker to arrange the cards without the players knowing about it? A magician could do it easily – by switching the pack for another while using a diversion to distract the attention of the players. It would be even easier if one of the players were themselves the practical joker. And when the coincidence crops up on 1 April at a gentlemen’s club, the chance that some underhand activity was behind it strikes me as really quite high.
There is something of a paradox here. The more unlikely a coincidence is, the more reason we have to believe that all is not as it seems. Imagine tossing a coin and getting heads 10 times in a row. That would be surprising and a bit unnerving. The chances of it happening are:
1/2 × 1/2 × 1/2 × 1/2 × 1/2 × 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = (1/2)10 h 1 in 1,000.
But suppose you keep flipping that coin and it comes up heads another 90 times, meaning that you have now got 100 heads in a row. The chance of this happening at random is ½100, or one in a million trillion trillion. What’s the chance that the next toss will be a head? Standard probability theory will tell you that the chance of another head is still ½. But the astronomically remote odds of you getting 100 heads in a row up to now means that other scenarios come in to play. What’s the chance that this coin is actually double-headed? Or that you always flip a coin identically so that it lands the same way up? Or that you have been hypnotised to believe that the coin is showing heads even when it isn’t? All of these are unlikely, yet they are far more likely than the chance that it is a fair toss with a fair coin.
Put it this way – if I were to toss a coin 100 times and get heads every time, and you asked me: ‘What are the chances that the next toss will also be a head?’ my answer would be: ‘It is almost certain.’
ENERGY, CLIMATE AND THE ENVIRONMENT
The future of the planet and how we treat it is one of the most pressing concerns of modern life. Nobody can be certain what the impact of climate change is going to be, as can be seen from the wide range of forecasts that come from the experts who build sophisticated computer models to predict this very thing.
The solutions on offer are consistent, however: reduce carbon dioxide and methane emissions (partly by saving energy), reduce the amount of waste that we produce, and for the waste that we can’t avoid, try to recycle as much of it as we can. Back-of-envelope calculations can help us to get a sense of both the scale of the problems, and the priorities for how to fix them.
WHAT USES THE MOST ENERGY AROUND THE HOUSE?
We’re being urged to do our bit to reduce global warming by saving energy. So it’s time to think about making some personal cutbacks.
Imagine you’re living alone in an apartment. You have a well-stocked fridge, a large TV on standby, you take a three-minute shower every morning, and you boil the kettle four times a day for coffee, tea and other essentials.
Which of these do you reckon is using the most energy over 24 hours?
(a) The fridge
(b) The TV on standby
(c) The shower
(d) The kettle
Among most audiences, there’s one answer that tends to be more popular than the others: (b), the TV on standby. There are probably two different reasons why people choose this. The first is a recollection of hearing once that TVs on standby use a lot more electricity than you’d expect. The other reason people choose the TV is to second-guess the questioner (‘the answer is going to be a surprise, so I’ll choose that one’).
In fact, the only one of the four options that is definitely not the biggest energy user is the TV on standby. Many years ago TVs on standby were indeed quite energy hungry (the TV would get quite warm, which is where the energy went) but those days are gone. A TV on standby now typically uses 1 to 2 watts, a fraction of the power of a regular lightbulb.
As to the biggest energy user, well – it depends.
A regular fridge (with no freezer) typically uses about 50W of power, not dissimilar to a normal lightbulb, but that does depend on how hard it is having to work (more energy is needed on hot days) and how efficient it is. It probably only has to work that hard for about half of the day, so that’s:
50W × 12 hours
= 600 watt hours.
~ 1/2 kWh in a day.
A typical kettle uses 2kW. If it takes three minutes (a twentieth of an hour) to boil the kettle, that’s 1/10 kWh per boiled kettle, so if we boil it four times, that’s:
But if the kettle is full, it might take a lot longer to boil, and we could easily get up to over 1/2 kWh.
And the shower? Suppose you boiled four kettles and put that in a water tank, and topped it up with water from the cold tap so that the temperature was hot but not boiling. How long would that last you as a shower. A minute or two, perhaps? So the energy need for that hot shower is not far off the amount needed for the four boiled kettles.
Depending on how hot the weather is, how full the kettle is and how long the shower takes, any of the three appliances could end up being the biggest consumer. In terms of orders of magnitude, they can all be treated as the same.
But there’s another everyday ‘appliance’ that on a typical day can consume energy at a higher order of magnitude. The car. If you start up your car and drive around town at a typical speed of 20 to 30 mph, stopping and starting at traffic lights, your rate of consumption of energy (or power) will typically average about 20kW. In other words, driving your car is roughly the equivalent of switching on 10 kettles, and leaving them boiling for the entire duration of your journey. A 30-minute car journey consumes more energy than all your domestic appliances put together. That’s something to think about when doing the school run. And as for that flight to Ibiza …
HOW MANY KETTLES IN A LIFETIME?
It’s sobering to think about how disposable so many of the products we use every day have become. It’s at least 10 years since two repair shops close to where I live shut down (one fixed televisions, the other fixed hoovers). Nowadays, when a TV or a vacuum cleaner stops working, we just chuck it out. And the sheer volume of ‘stuff’ that all of us might chuck out per year is mind-boggling.
Let’s just pick one innocent household item as an example: the kettle that we looked at in the previous section. When I was a boy, we had an aluminium kettle that sat on the gas hob. We used that when making tea, and it lasted my entire childhood. But since I left home, I’ve always had an electric kettle, as most of us do. And kettles are the sort of thing you might expect to just be ‘there’ for a whole lifetime. But how many kettles is that likely to be?
I’ve been married for 20 years, and a kettle was one of our wedding gifts (do people still give kettles as wedding gifts?). Sadly, and please don’t tell whoever gave it to us, that kettle only lasted
a couple of years. We are now on our fifth kettle since the first one stopped working. Have we been unlucky, or does this mean that a typical kettle now lasts only three or four years? If we assume kettles are something adults are responsible for (and children are just beneficiaries of the kettles that are there), then it looks like somebody living to 80 might easily be getting through 20 kettles in their adult life. OK, so households usually have more than one adult, so that might be more like 20 kettles per household rather than per person, but still, my grandmother’s generation would be staggered if they could see the rate at which we get through appliances that used to last most of a lifetime.
As a society we have become used to this pattern of short-lived, disposable household goods in the last 30 years or so, and it’s now hard to imagine how life was beforehand. But let’s look ahead 100 years. At this rate of disposal …
30 million households in the UK
100 years ÷ 4 years per kettle = 25 kettles per household
30 million × 25 = 750 million kettles
… we will have got through close to one billion more kettles by then. In the UK.
Suppose this does happen. If we’re lucky, the metal elements from most of them will have been recycled, but the rest will be filling landfill sites somewhere.
Is this sustainable? Surely not. And what about in 500 years’ time? It’s hard for us to imagine what life will be like by then, but long beforehand there is going to have to be a seismic shift in our lifestyles and our consumption.
HOW MUCH FRESH WATER IS FLUSHED DOWN LONDON TOILETS EVERY DAY?