Meantime, Here is the News from Greenwich

Hopperational details
Date & Venue	Saturday 24 February 2018 at The DGS Stadium.  Cray Valley (PM) are groundsharers here.
Result	Greenwich Borough 1 Chipstead 0
Competition	Isthmian League Division One South (Step 4)
Hopstats	Ground 636 on the lifetime list.  Not enough time for a long Step 3 drive today so chose my nearest Step 4 unvisited ground and made it through the Blackwell Tunnel in time.
Context
Greenwich Borough have been in the top six for most of the season but arrive at today with a run of two defeats. Chipstead are mid-table but have won their last three, so it should be a competitive game.
In one sentence
Not a classic on a bright and cold day, but Greenwich breached a stubborn Chipstead defence with a scruffy late goal to take the points.
So what?
The teams finish the day in 5th and 14th place respectively.
Match Report
There was not much to write about in the first quarter.  On a very cold but bright day, Chipstead started with the wind and down the slope, and too many passes were overhit.  They were more direct and through-the-middle than the home side, who used their wide players more.  However, Chipstead’s defence was strong and well-organised.  Other than a two-on-two which Greenwich failed to use, the first half-hour passed without much goalmouth action.  Chipstead conceded (and complained about) a number of free-kicks in midfield, and again a good deal of overt dissent was ignored by the referee. On 32 minutes, Borough’s Jack Barham showed great skill in controlling a flick-on, then turning and volleying to bring a good save from Chipstead keeper Michael Sibley.  However, Chipstead had their best two chances before the interval.  Michael Campbell latched on to a through ball and lobbed the ball over the advancing keeper Michael McEntegart, but also just over the bar.  Then Sam Bell headed wide, presumably not realising that Sam Page was better placed just behind him.  So, goalless at the half-way point. In the second half, Greenwich began to build the pressure, but the Chipstead defenders were more often than not able to protect their keeper by blocking shots at source.  The home side were running with the ball rather than playing passes into the channels.  Sibley made a great save to deny Barham and it looked as if Chipstead had weathered the storm.  Their own occasional breaks were snuffed out easily enough but it looked like the game was heading for a draw. With seven minutes of normal time, Greenwich substitute Omari Delgardo-Hibbert showed strength and perseverance to get the ball under control near the byeline.  He smashed the ball very hard into the six-yard box and it was deflected over the line at the near post.  They all count, and this one settled the game.  Three hard-earned points for Borough and no reward for a hard afternoon for the Chipstead defence.
Ground Pix
The ground is a work-in-progress following Greenwich Borough’s arrival this season.  It is now predominantly branded in the club’s red-and-black though Cray Valley (PM) remain as the other groundsharers.  Some very deep potholes at the entrance to the car park so groundhoppers with Ferraris or Lamborghinis are advised to use their mopeds instead.
Match Pix
Greenwich Borough in red-and-black.
Goalkeeper Top Colour Stats
Blue beats Yellow with a clean sheet too and jumps one place in the table.  Michael McEntegart is (as far as I know) the first keeper I have seen twice in one season wearing different colours.  Last month he was in pink for Ashford United at Faversham, and he must have joined his manager Paul Barnes in making this recent move. Results so far: Based on conventional 3pts for a win, 1pt for a draw, but also -1pt for a goal conceded (GC) and +5pts for a clean sheet (CS).  Colours ranked on a points per game (PPG) basis. For new readers the odd .5 was caused by a shocking half-and-half shirt and the .1 was due to a substitute goalkeeper in a different colour.  The Fire Cracker colour was confirmed with the help of the social media team at Dulux UK.  All of this arises from a comment attributed to Petr Cech that orange is the best colour for a goalkeeper because it changes the behaviour of other players around the box. P W D L GC CS Pts PPG Red 10.0 5.0 1.0 4.0 11.0 3.0 20.0 2.000 Fire Cracker 2.0 1.0 0.0 1.0 4.0 1.0 4.0 2.000 Blue 32.1 14.0 6.0 12.1 49.0 11.0 54.0 1.682 Maroon 4.0 2.0 1.0 1.0 6.0 1.0 6.0 1.500 Green 73.0 36.0 10.0 27.0 122.0 19.0 91.0 1.247 Grey 43.5 20.0 9.0 14.5 72.5 11.0 51.5 1.184 Orange 29.5 9.0 6.0 14.5 49.5 5.0 8.5 0.288 Radioactive Bile 13.0 6.0 0.0 7.0 26.0 2.0 2.0 0.154 Yellow 31.0 8.0 7.0 16.0 65.0 5.0 -9.0 -0.290 Purple 12.0 5.0 2.0 5.0 31.0 2.0 -4.0 -0.333 Pink 15.0 4.0 5.0 6.0 30.0 1.0 -8.0 -0.533 Black 5.0 1.0 3.0 1.0 14.0 0.0 -8.0 -1.600 White 1.9 0.0 0.0 1.9 4.0 0.0 -4.0 -2.105
What Next?
Follow @GrahamYapp on Twitter for details!  I am back to science teaching three days a week again for the foreseeable future, so may have to defer some of the longer journeys until later in the season.

I Could Have Chanced All Night

After the quarter-final draw for the FA Cup had been made, I stumbled across comments on Twitter about the hot ball-cold ball theories of draw-fixing.  The “top four” teams in the competition all avoided each other.  The best argument against fixing comes from the fact that different combinations of players, ex-players and celebrities are used by the media in each round.  When you consider how many people by now would have to be willingly complicit in the arrangements, and to have kept their silence all these years, the idea becomes very, very implausible.  Nevertheless there are some people who take a lot of convincing, so let’s look at the maths.  Credit to my son Chris who took a look at my first attempt, helped me spot a reasoning error and gave me an idea for a better way of presenting the arguments visually.

What are the chances of the top four teams in a QF draw all avoiding each other?

Let’s imagine a completely fair draw in which the best four teams are given RED balls and the others are BLUE.  We don’t care about the home-and-away order, just the pairings.  We start with eight balls in the bag – and so there are eight possibilities for the first ball out.  Then there are seven different possibilities for the second ball drawn out, so even at this point there are 8x7=56 different possible outcomes.

There are 6 alternatives for the third ball, and so on, so the total number of different draw sequences for the eight balls is 8x7x6x5x4x3x2x1, known as 8! or “factorial-8”, and this has a value of 40,320.  We will need this number later on.

Now, let’s go back to the start of the draw, and the first two teams drawn.  The first could be either a red (top four) or blue (not top four), and likewise the second.  But the relative chances for the second ball will depend on what happened for the first.  The chances are shown in the following table.  The first ball has equal (4 out of 8) chances for red and blue.  Now there are only seven balls in the bag.  If the first one was red, then the chance of the second one being red is now 3 out of 7.  We are interested in the chances of getting one blue and one red, in either order.  The combined chance is calculated by multiplying together the chances in the first ball and second ball columns, and we end up with a number out of 56.  The first tie would be red v red (top four v top four) on 12 out of 56 pairings, red v blue on 32, and blue v blue on 12.

First Ball Out	Second Ball Out	Outcome	First Tie Combined Chance
Red (4/8)	Red (3/7)	Red v Red	12/56
	Blue (4/7)	Red v Blue	16/56
Blue (4/8)	Red (4/7)	Blue v Red	16/56
	Blue (3/7)	Blue v Blue	12/56

Notice that the sum total of all the combined chances is 56/56 – which must be true because there are no other possibilities.  The chance of one top team and one lower team – in other words one blue and one red in either order – is 16/56 plus 16/56, which is 32/56 or 4/7.  This means that 4 out of 7 of the 40,320 possible draws would have pitched a top team against a lower team.  The number is 23,040.  We are no longer interested in the other 17,280 possibilities because the first tie out of the bag would have brought up either two top teams (red) or two lower teams (blue) to play each other.  If the latter, then it would be guaranteed that at least one of the other three ties would be red v red.  So only 23,040 or the 40,320 possibilities keep the “perfect separation of the top four teams” alive as a possibility.

Let’s assume that first tie was either red v blue or blue v red, and we go on to draw two more balls for the second tie.  Again, we are interested in the case where we again draw a red and a blue, in either order.  However, this time we start with only six balls left in the bag (three red, three blue) so the fractions are different, as shown here.   Same logic, just different values for the fractional chances.

Third Ball Out	Fourth Ball Out	Outcome	Second Tie Combined Chance
Red (3/6)	Red (2/5)	Red v Red	6/30
	Blue (3/5)	Red v Blue	9/30
Blue (3/6)	Red (3/5)	Blue v Red	9/30
	Blue (2/5)	Blue v Blue	6/30

Again, the total of the combined chances adds up to a certainty, or 30/30.  The red-blue or blue-red combination that interests us occurs in 18/30, or 3/5 of the 23,040 draws we are still considering out of the possible 43,020.  We are therefore now down to 13,824.

Let us now assume that BOTH of the first two ties have been red-blue or blue-red, and go on.  Now there two red and two blue in the bag, and the next table shows the possible outcomes for the 5^th and 6^th balls drawn.

Fifth Ball Out	Sixth Ball Out	Outcome	Third Tie Combined Chance
Red (2/4)	Red (1/3)	Red v Red	2/12
	Blue (2/3)	Red v Blue	4/12
Blue (2/4)	Red (2/3)	Blue v Red	4/12
	Blue (1/3)	Blue v Blue	2/12

The chances of another red-blue combo for the third match is 8/12 or 2/3 of the 13,824 we were still considering of the original 40,320, and that is 9,216.  There is nothing else to consider for the fourth match – if the first three are red-blue pairings, then the fourth match is bound to be the same.

So, it can be seen that in a perfectly fair 8-ball quarter final cup draw,  9,216 of the 40,320 possible draw sequences would keep the “top 4” teams in the draw away from each other.

That’s 22.9%.

I’d bet that would be in line with historical analysis if anyone has the time!

The same maths would apply to any arbitrary four : four split of the eight teams.  For example, if there were four “northern” teams and four “southern” teams, the probability of all four ties being north v south confrontations would be the same 22.9%.  It would mean that the northern teams had avoided each other, and so had the southern teams.

All of this also explains why competent mathematicians are deeply interesting people and well worth your time ;)

Extra bit: 

Why are we sometimes multiplying fractions and sometimes adding them?

In simple terms, when we want to find the probability of two independent events happening, multiplication of the two chances is needed.  The key word is AND.  This applies when one event AND the second event are to happen independently.  So for a red ball to come out first of all (4/8) AND then followed by another red ball (now 3/7), then the calculation is 4/8 x 3/7 which is 12/56.

However, when we wanted to find the total chance of a red-blue OR a blue-red combination, the key word is OR.  We need to add together the two separate chances for the events.  Blue-red is 16/56 and so is red-blue.  16/56 + 16/56 = 32/56 which is 4/7.

Links to other Modus Hopper Random posts on cup draws:

Scottish Cup Semi-Finals and Old Firm Conspiracies

http://modushopperrandom.blogspot.co.uk/2010/10/controversy-in-morning-fourballs.html

Champions League Quarter Final Draws

https://modushopperrandom.blogspot.co.uk/2011/03/embrace-numbers-quarter-final-draws.html

Updated Version

https://modushopperrandom.blogspot.co.uk/2014/03/crunching-numbers.html

Links to Modus Hopper Random posts on football games involving dice and chance:

Wembley

http://modushopperrandom.blogspot.co.uk/2011/06/diceman-goes-to-wembley.html

Soccerama

http://modushopperrandom.blogspot.co.uk/2011/06/putting-drama-in-soccerama.html

Soccerboss

https://modushopperrandom.blogspot.co.uk/2013/07/soccerboss-living-dream.html