Tangential Ramblings

What are the odds?

dan — 11:46pm, Saturday 11 October 2008

Tonight's winning lottery numbers were 20, 21, 23, 24, 27, 28. The chances of all numbers appearing in the same set of ten are 0.0066%, or 1 in 15,134. One lottery every 145 years. The likelihood of them all being in their twenties is 0.0015%, or 1 in 66,590. One lottery every 638 years.

Of course, the likelihoodof those specific numbers coming up is 0.00000715%, once every 19,142 years.

It could be you!

World's smallest prime

dan — 10:30pm, Tuesday 30 September 2008

While US mathematicians are busy confirming their finding of the largest known prime, over 13m digits in length, I can proudly reveal that I have discovered the world's smallest prime.

I've been working with over 100 other computer owners over the last seven months. Together we've whirred through the numbers checking every one, and have discovered the smallest: 2.

Not only is it the smallest known prime, initial tests suggest that there aren't any smaller to be found.

I'm surprised to discover that the Electronic Frontier Foundation is not offering a prize for our discovery. Bugger.

Poker justice

dan — 9:53pm, Friday 26 September 2008

I played poker last night. Texas hold 'em, apparently. The hand you bet against is the best five-card hand you can muster from two cards dealt to you and five that are dealt face-up on the table.

You're dealt two cards each. Then there's a round of betting. Then three cards are drawn, all face up on the table. Another round of betting. Card four on the table. Betting. Card five. And final betting.

�10 was the initial stake, with unlimited buy-ins before 10pm, after which, you bet until you won or you were eliminated.

Just before midnight, the nine players had been whittled down to two, of which I was lucky enough to be one. The �200-worth of chips (and supplementary hand-written 100-chip notes) were pretty evenly divided between the two of us, so instead of battling through the night we thought it wise to split the pot. �80 up on the night: not bad at all.

The reason for the post, however, is to question one specific feature of the game. If a player runs out of chips during betting, they can still stay in for that round, not being obliged to bet thereafter. If they win the round, then their winnings from each other player are capped at their own bet. So if Alberto has bet his last 25 chips, Balthazar 70 (before folding) and Cecilia 90 (still in), then Alberto winning means that they will take 25 from each of the other two players.

At first glance this seems quite fair. But the scenario has always troubled me, and I've just realised why. Let's assume that the three players' hands have equal chances of winning, and it is merely the playing styles of the players that are at odds.

If Balthazar and Cecilia get into a bidding war after Alberto has maxed out, Balthazar folding when Cecilia raises that bit too much to justify him continuing, then the increased investment by Balthazar and Cecilia has benefited Alberto, as he now only has to beat one hand rather than two. So Alberto has benefited from the additional investment of Cecilia, through it driving Balthazar out of the game.

Further, continued investment from one of the cash-rich players cannot result in them winning by default�they will always have to beat Alberto to win the round.

The trade-off is that Alberto can win fewer chips than Balthazar or Cecilia. But there is no trade-off for Balthazar or Cecilia.

Premier League predictions

dan — 7:49am, Sunday 14 September 2008

If at the start of the season you were to predict the finishing positions of the 20 Premier League teams and then compare your predictions with their actual finishing positions, if you had no previous knowledge of teams' performance, what would be your expected margin of error? The measure here is the sum of the absolute differences between teams' predicted and actual positions. So if every team was out by one (either over- or under-predicted), then it would be 20.

I'm not sure whether it's easy to create a formula for n teams, but a random Excel trial of 26,126 such prediction sets yielded a minimum difference of 56 (an average discrepancy of 2.8 positions per team), a maximum of 192 (9.6 positions per team) and an average of 131 (6.6).

My brother's performance in his work competition based on the teams' current positions is 80, 0.4% of my random trials bettering this. Is 80 good? And where will he be in May?

Pace by distance

dan — 8:40pm, Monday 25 August 2008

I read an article today on the Freakonomics blog about the relative paces of world record-holding athletes over different distances. Its specific focus was on who was indeed the fastest person in the world, and whether the 100m or the 200m world record holder was, on average, faster. (The point is somewhat moot now that Usain Bolt holds both records.) Anyway, I did some analysis.

Below is a chart I put together showing the average speed of different distances' world record holders over 100m.

Right-click and View Image for a closer look at the chart in FireFox; there doesn't seem to be an equivalent in IE7. I've given cuts of the data every ten years, working back from 2008. For each series, data starts where records begin. The half-marathon has the steepest gradient in recent history, probably down to the event being taken more seriously in more recent years. (For completeness, the 1908 marathon record equated to 24.93 seconds per 100 metres, but its inclusion squashed the scale unnecessarily.) The 60m record is 9.9% slower in pace than the 100m record, but the 100m and 200m paces have pretty much kept in line with one another, the extra distance and the impact of the bend in the 200m being counterbalanced by the impact the acceleration has on the time of the 100m. Below is a closer look at these two.

No conclusions; merely food for thought. Maybe.

Full Olympic medal analysis

dan — 2:56pm, Sunday 24 August 2008

So, the Olympics are over. And here�s a full analysis of the 958 medals won.

In terms of the traditional measure of number of gold medals won, China take the honours with 51, followed by the USA (36), Russia (23) and Great Britain (19). The same order is maintained for the top four if you base it on a scoring method of three for a gold, two for a silver and one for a bronze, China scoring 223 points, USA 220, Russia 72 and Great Britain 47. If you forget the medals� colour and base it purely on the medal count, then the USA (110) and China (100) switch places, Russia (72) and Great Britain (47) remaining in third and fourth respectively.

If instead you look at medals compared to countries� populations, then it�s a different story. The Bahamas� two medals equate to one for every 165,500 people. Jamaica come second (eleven medals; one per 246,727 people); Iceland third (one medal; one per 316,252 people), with the top five rounded off with Slovenia (five medals; one per 405,800) and Australia (46 medals; one per 444,221). The least successful of the 87 medal-winning countries by this measure were India (three medals, one per 376,622,051), Vietnam (one; 85,262,356), Egypt (one; 75,231,000), South Africa (one; 47,850,700) and Indonesia (five; 46,938,799).

Excluding relatively trivial examples (denominators less than seven), the most successful countries in terms of percentage golds were Ethiopia (57%; four of seven), Jamaica (55%; six of eleven), China (51%; 51 of 100), Romania (four of eight), the Netherlands (44%; seven of 16), South Korea (42%; 13 of 31) and Great Britain (42%; 19 of 47).

Lightning Bolt

dan — 9:21pm, Thursday 21 August 2008

A phenomenal graphic courtesy of the New York Times' Freakonomics blog showing the 252 fastest 200m sprints of all time.

Here is the fully interactive version.

Bolt and Johnson are outliers to a ludicrous degree. There are three whole tenths of a second devoid of any times�from 19.32s to 19.62s. If you except Johnson's previous world record time, Bolt is 1.63% quicker than the next best time ever (Tyson Gay's last year). Over such a short distance, that's remarkable.

When extrapolated to the marathon, it equates to a margin of 2m 4s compared to Haile Gebrselassie's time of 2h 4m 26s in Berlin in 2007. Three athletes have managed that feat, Khalid Khannouchi doing so twice. On the women's side, Paula Radcliffe's 2003 London marathon (2h 15m 25s) was 2.43% quicker than the next fastest, Catherine Ndereba's 2001 Chicago effort. Again, a ludicrous margin and a record that will surely be tough to beat.

Interestingly, the late Florence Griffith-Joyner's 200m time of 21.34 during 1988's Seoul Olympics is 1.39% quicker than her fastest rival, Marion Jones' 21.62 recorded in 1998 allegedly before her drug phase. Whether indeed either of these times were drug-assisted will never be known.