Monday, 1 September 2008

Unlucky, Captain?

As England, have, remarkably enough, won five games in a row under new skipper Kevin Pieterson, there has been some debate on the traditional question of whether KP is a good captain or a lucky one...I’m not going to consider this here, but it did remind me of the time in 2000-2001 when then England captain Nasser Hussein somehow contrived to lose 14 tosses in a row. Surely you can’t get more unlucky than that? Maybe there’s even something spooky going on? (Some non-mathematically minded commentators even suggested he should stop calling “heads” each time to give him more chance of finally winning a toss.)

But. The chances of losing 14 tosses in a row is 1 in 2^14, or 1 in 16,384. This is indeed fairly astronomical. However. Hussein captained England in 45 Tests and 56 ODIs, meaning he contested 101 international tosses in total. There are, unless I’ve made an egregious blunder, 88 possible starting points for a sub-sequence of 14 in an overall sequence of 101, so the chance, sometime in his captaincy, of Hussein losing 14 tosses in a row is actually something of the order of 1 in 186 or thereabouts, which isn’t nearly so unlikely...

Mind you.
If not exactly a case for Mulder and Scully, was this still, nevertheless, slightly odd and unexpected? How often might we expect a sequence of x consecutive heads or tails in a particular sequence of tosses? The chance of, say, x tails from a given start point is (½)^x. If there are N possible starting points for the sequence, the average number of times a sequence of x tails appear will be N x ((½)^x). So, with 200 possible start points for a sequence of 5 tails, we would expect, on average, about 200 x ((½)^5), or about 6, such sequences to turn up.

So. Let’s say we decide (as seems quite reasonable), that we wouldn’t be surprised at a particular sequence occurring, if, on average, we would expect at least one such sequence to occur; in other words when N x (½)^x is as least 1.

The expression will equal 1 when x = Log (base 2) N.

E.g. the base 2 log of 64 is 6, and 64 x ((½)^6) = 1, and the base 2 log of 512 is 9, and 512 x ((½)^9) = 1.

So, with 88 possible starting points for a sequence, the expression will equal one when x = 6.459, i.e. 88 x ((½)^6.459) = 1.000. In other words, a total lack of surprise at Hussein’s tossing performance (as it were), would only be preserved if he had lost about 7 tosses in a row, rather than 14.

(From the above, we can use the elegant, if admittedly approximate and somewhat arbitrary “rule”, that if we toss a coin N times, we should expect the longest sequence of heads or tails to be round about the base 2 logarithm of N. The base 2 log of 101 is about six and two-thirds, give or take.)

There we are then. Although not colossally surprising (especially given that Hussein is not the only international captain who has taken part in a great many coin-tosses, so before being amazed we need to think beyond the chances of just him having such a run of bad luck), it was indeed perhaps a slightly odd run. Not nearly as odd as England being 4-0 up in a one-day series against South Africa, though. Funny old game, isn’t it?

Due Credit Department: [For more of this sort of thing, see Beating the Odds by Rob Eastaway and John Haigh (Robson Books, 2007)]

14 comments:

Lee said...

I liked that... thanks.

Sometimes the odds seem 'amazing' until someone does the stats – so I am glad you did.

Not only are there many international captains doing the coin toss each week - it has been going on for many a long year.

In 8 years since this Hussein 'fluke' how many coin tosses has there been in international cricket (and should we also include county cricket since I am sure this would have got a mention by the commentators should 14 in a row have been reached) Also, hasn’t international one day cricket been going on since the 70’s?

Are we getting closer to a certainty yet? It was bound to happen one day?

Don't know - I'm rubbish at stats but million to one events happen everyday, this I am sure. Someone certainly buys my winner lottery ticket since it isn’t me.

For more of this sort of thing, see Beating the Odds by Rob Eastaway and John Haigh

Erm... sounds interesting. I'll write a letter to Father Christmas at once.

Lee

Billy said...

I wonder what the probability of us (the individuals, not the species) arising just after the big bang was? It does show that something improbable is possible

Mark_W said...

Lee,

Cheers - and you're right: as the old saying goes, coincidence is often far less unlikely than we think...(The famous old probability chestnut that still amazes some people is that you need to gather together a group of only 23 people before there is a better than fifty-fifty chance that two of them share a birthday...)

And yes, given the amount of cricket played (especially nowadays), and looking beyond just one captain's results, the chances that someone would, at some stage, lose that many tosses are not really that surprising...

Billy,

I wonder what the probability of us (the individuals, not the species) arising just after the big bang was?

Indeed. I'll hopefully come back to questions like this: it was your previous comments on the way McGrath tries to use probability that got me thinking about these sorts of things again - many years ago now I did a statistics degree, but I haven't really used it for over a decade and a half now, so wanted to flex the old maths muscles a bit with something like cricket before moving on to the harder stuff! I'll hopefully come back to this, though...

Mark_W

Lee said...

Billy asked I wonder what the probability of us (the individuals, not the species) arising just after the big bang was?

I think 10 to the power of gigazillion to one? Probably higher
(I made that up of course)

The number (if you could work it out) would be stupidly high as you know so high as bloody unlikely.

However, we are here... end of story

Thing is, for me, it would be asking the wrong question – any number given (and theist’s do throw some around from time to time for the species) is meaningless.

Since we are not asking what are the changes of it happening twice?

Example?

Get a bucket of red paint - go to a bridge or some other high place - and throw the paint over the side to try and hit a very large piece of paper on the ground (that you placed there earlier)

You should get a pretty red pattern...

Now, what are the changes of making the pattern you have just created?

Well, the changes were close to 1 isn’t it? It was going to happen as soon as you dropped the paint towards the paper.

However, what are the changes of doing the same pattern again by just throwing paint from a high building – it is highly unlikely you could repeat it.

So this is why I think any probability comparisons to a single event are silly and meaningless.

And also, without any comparison (i.e. what are the chances of God deciding to do something) any odds a natural explanation has will be better than nothing.

Mark_W many years ago now I did a statistics degree, but I haven't really used it for over a decade and a half now, so wanted to flex the old maths muscles a bit with something like cricket before moving on to the harder stuff! I'll hopefully come back to this, though...

I encourage you to play with more statistics... it will be interesting.

...and erm, I hope my comment above to Billy is correct – I will bow down to your statistics degree if you wish to correct me.

Lee

Billy said...

Hi Lee I think if you re ran the big bang, you almost certainly would not get us appearing. It would require all the chance events to re run exactly. If we could go back 500 million years and re run evolutionary history, it would be unlikeley to be here - that important ancestor may have turned left instead of right and got eaten instead. The thing with Mcgrath is that he uses things like this as an example of something unlikely not being impossible. However, this is not the same as asking the question how likely is something (that is not as obvious as you or I are) exist given what we know. We know that we exist, that's where Mcgrath screws up, so given what we know, the probability that we exist is 1.

Mark_W said...

Lee and Billy,

However, we are here... end of story

Indeed. At the moment I am listening to Bruce Springsteen's Magic album on random play. (An experience I'd recommend, as Gypsy Biker is the best song Springsteen (or anyone, really) has written in the last 20 years...)

Anyway, the chances of the particular order the 12 tracks appear in tonight occurring will be ridiculous, but the chance that one of the possible sequences will occur (barring the end of the world in the next twenty minutes) is 1, so the catastropic unlikelihood is only meaningful if the order is specified beforehand...

Naturally, with anything interesting like the apparent fine tuning of the universe, things aren't always this simple - but I'll come back to this next week, hopefully (Dad duties this weekend!)

Mark_W

Lee said...

The thing with Mcgrath is that he uses things like this as an example of something unlikely not being impossible. However, this is not the same as asking the question how likely is something (that is not as obvious as you or I are) exist given what we know. We know that we exist, that's where Mcgrath screws up, so given what we know, the probability that we exist is 1.

It’s the fear of large numbers McGrath is playing on... but yes, we exist – the probably is 1. Anything else McGrath does with the numbers are pretty meaningless.

If however you wish to place a number on the chances of it happening again then that is fine... the number will be huge, but I am happy with just the one universe and the one me (two me’s the world just doesn’t want to see)


At the moment I am listening to Bruce Springsteen's Magic album on random play.

I won’t judge you because of this :-)

Naturally, with anything interesting like the apparent fine tuning of the universe, things aren't always this simple - but I'll come back to this next week,

I look forward to it.

(Dad duties this weekend!)

Father’s Day this Sunday for me.

Excellent

Lee

Mark_W said...

Lee,

At the moment I am listening to Bruce Springsteen's Magic album on random play.

I won’t judge you because of this :-)


Awfully decent of you, Sir. :-) And have a splendid Father's Day...

Mark_W

Lee said...

And have a splendid Father's Day...

Oh I did thanks... cup of tea in bed, fried breakfast, BBQ and a homemade card from my eldest (which I am told had a lot of stars on it - maybe shooting stars because it was mainly lines to my eyes)

Perfect.

Lee

][vellios said...

Amazingly, Every Time He Goes For The Coin Toss His Chances Are Still 1 In 2. He's Got A 50/50 Chance.

In This Link It Points Out That The Odds Only Count Before The First Toss.

http://en.wikipedia.org/wiki/Gambler%27s_fallacy


"While a run of five heads is only 1 in 32 (0.03125), it is 1 in 32 before the coin is first tossed. After the first four tosses the results are no longer unknown, so they do not count."

Mark_W said...

][vellios

It’s true, as per The Gambler’s Fallacy, that the probability of Hussein losing 14 tosses in a row, given that he has already lost 13 in a row is indeed 1 in 2, but this is not what we were considering here – rather it was the probability that in a sequence of 101 tosses a particular run of 14 would occur, as compared with the probability of an individual sequence of 14 happening in isolation.

It’s correct that once Hussein had already lost 13 tosses, no-one should have been at all surprised that the run of bad luck extended to 14, since, as you say, each toss is independent and, at this point, he only had a fifty-fifty chance of ending the run and finally winning one...

However, we were considering the probability of whole runs of (in this case of 14), and, more generally, the likelihood of particular sequences occurring, and were not starting from a position of some of the results already having occurred...

Mark_W

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