For those outside Australia, or for those Australians who are living (or, understandably, hiding) under a rock, we've just had our national elections, at which our all of the seats of our government have been decided and half of the seats in our Senate (the house of review).
Though almost all of the seats in the lower house have been decided, which is normal for election night, the results for the Senate generally take days to weeks to be fully finalised. Though most of the seats are generally worked out fairly quickly - in particular, those seats going to the major parties - the remaining few seats are far less certain.
The use of the Single Transferable Vote system for the Australian Senate means that votes for minor parties go through a convoluted process of 'transfer' from candidate to candidate, which is further complicated by the Group Voting Ticket system and the deals made by minor parties with each other for preferences. What this means is that a party receiving a very small number of votes can obtain a seat in the Senate simply by the snowballing of preferences from other small parties.
This has been particularly apparent in this election, with the current estimated results by the ABC suggesting that as many as 8 seats are likely to go to parties outside of the main three (the Liberal/National coalition, the Australian Labor Party and the Australian Greens), with seats controversially likely to go to members from the Australian Sports Party and Australian Motoring Enthusiasts Party, which only received a tiny fraction of the initial vote. The popular media has already heavily covered these results even though they are still by no means yet certain.
Because of the above complexities, it can take only a small variation in voting to change the result for one or more seats. In this sense, the ABC's estimate is fairly naive: they assume that all voters have voted 'above the line', allowing their preferences to be decided by their chosen party (though this is not so far from the truth, with over 95% of voters generally doing so) and that the final results will be accurately represented by the results that have come in so far (between 50-80% of the vote for each state). Working out what potential bias there may be in the remaining votes is possible to a certain extent, as the voting information includes voting breakdowns for smaller regions (and can be compared with past elections), and some regions are known to have regular skews in their voting patterns.
What I've done here more simply, however, is to look at how much effect there might be in random fluctuations in the remaining votes to be counted. I assumed that the proportions of votes to each party so far were an accurate representation of the electorate's intent - based on those numbers, I randomly generated the remaining expected votes to be counted (based on current enrolment numbers and last election's turnout - around 94% on average).
For Tasmania, for example, my results usually follow the ABC's results - two each of Labor and Liberal senators are elected, one Greens senator, and one from the Palmer United Party are elected as expected. However, in about 4% of cases (for 1000 election runs) a member of the Sex Party is elected instead of the Palmer United candidate, and in a further 1% of cases a third Liberal Party member is elected.
Taking into account the other sources of fluctuation mentioned above adds to this uncertainty in the results - the Geeklections site and the Truth Seeker blog go into much more detail. This only goes to show that surprises are not only possible but likely as the counting continues...
Monday, September 9, 2013
Monday, August 12, 2013
Why we can't really see the stars
If you're like me, you enjoy looking up at the stars at night and thinking about how far away they are, and such things. Recently, though, I started wondering why there aren't any high quality images of stars other than our sun. The star with the largest apparent size from Earth (after the sun, again) is currently believed to be R Doradus - and the photograph of that on Wikipedia isn't exactly spectacular:
I don't know anything much about astronomy so this seemed strange to me. If I can see the stars with my naked eye, what's to stop someone with a high powered telescope zooming in and getting good details?
The reason, as I found out, is that stars are much, much further away than they look when viewed with the eye. The main reason for this is that every lens, including the human eye, has a limit to the resolution it can see. This is known as the 'diffraction limit' because once light travels through an aperture (in our case, our pupils), the waves spread out before hitting the detector (our retinas), blurring each point into what is called an Airy disk. For a human with 20/20 vision, the Airy disk is about an arcminute in size - so our sight can resolve something 1 inch in diameter from about 90 metres away. Every star we see looks 'blurred' to about this size - which is why all stars in the sky (except, once more, for the sun) look the same size.
To be able to escape the diffraction limit, we need a much larger lens - which is why we use telescopes. However, once a telescope reaches about 10cm in diameter, another effect stops us from seeing the star - a phenomenon known as 'astronomical seeing'. This is the effect caused by variations in temperature and wind speed in the atmosphere causing the light to bend on the way to the receiver. The 'twinkling' that can sometimes be seen in stars is due to this effect, as the apparent position of the star moves with the constantly changing conditions in the atmosphere.
At a good astronomical site, astronomical seeing will allow for a resolution of around 1 arcsecond. As illustrated above, this is roughly sixty times smaller (in blue) in length than human vision (in white) but even this is not enough to see a star. Below is the resolution with atmospheric seeing in blue again, but with R Doradus pictured in red - with a radius of 0.057 arcseconds. The only reason that ground-based telescopes are able to image R Doradus at all is by using adaptive optics - this attempts to compensate for the atmopsheric effects, and even this technology is currently only just enough to get a picture.
I don't know anything much about astronomy so this seemed strange to me. If I can see the stars with my naked eye, what's to stop someone with a high powered telescope zooming in and getting good details?
The reason, as I found out, is that stars are much, much further away than they look when viewed with the eye. The main reason for this is that every lens, including the human eye, has a limit to the resolution it can see. This is known as the 'diffraction limit' because once light travels through an aperture (in our case, our pupils), the waves spread out before hitting the detector (our retinas), blurring each point into what is called an Airy disk. For a human with 20/20 vision, the Airy disk is about an arcminute in size - so our sight can resolve something 1 inch in diameter from about 90 metres away. Every star we see looks 'blurred' to about this size - which is why all stars in the sky (except, once more, for the sun) look the same size.
To be able to escape the diffraction limit, we need a much larger lens - which is why we use telescopes. However, once a telescope reaches about 10cm in diameter, another effect stops us from seeing the star - a phenomenon known as 'astronomical seeing'. This is the effect caused by variations in temperature and wind speed in the atmosphere causing the light to bend on the way to the receiver. The 'twinkling' that can sometimes be seen in stars is due to this effect, as the apparent position of the star moves with the constantly changing conditions in the atmosphere.
At a good astronomical site, astronomical seeing will allow for a resolution of around 1 arcsecond. As illustrated above, this is roughly sixty times smaller (in blue) in length than human vision (in white) but even this is not enough to see a star. Below is the resolution with atmospheric seeing in blue again, but with R Doradus pictured in red - with a radius of 0.057 arcseconds. The only reason that ground-based telescopes are able to image R Doradus at all is by using adaptive optics - this attempts to compensate for the atmopsheric effects, and even this technology is currently only just enough to get a picture.
A large enough orbiting telescope would get past both of these effects - the Hubble Space Telescope is still one of the largest with a mirror 2.5 metres in diameter*, which translates to a 0.05 arcsecond resolution for visible light: only just enough to see R Doradus.
So humanity has a long way to go yet before we can really see the stars. Now if only I could afford a telescope...
* the largest, the Herschel Space Telescope, has a diameter of 3.5 metres.
Monday, July 22, 2013
Crappy days
Some days you just know are going to be long and painful. I have a few strategies to survive mine:
1. Sugary substances
Chocolate in any form is always appreciated, but on cold, miserable winter days a nice warming hot chocolate or Milo (link for those not in Milo-drinking countries) can make it all seem a little better.
1. Sugary substances
Chocolate in any form is always appreciated, but on cold, miserable winter days a nice warming hot chocolate or Milo (link for those not in Milo-drinking countries) can make it all seem a little better.
2. Cute things on the internet
It's an internet cliche because it works - my girlfriend (who now has a blog!) is usually my main source of such links. However, I always keep this one on standby for particularly bad days - it takes a cold soul indeed not to find this one cute:
3. Puzzles
When it's hard for me to concentrate on things I should be actually working on, I sometimes find doing some puzzles a good way to keep my brain ticking over. My current favourite is Project Euler (warning - non-programmers will really struggle!)
4. Music
I'm regularly surprised by how much music can help turn a mood around or focus the energies - I've never been much of an electronica fan, but iriXx's work has given me some of my most productive afternoons. I tend to listen to the same music over and over again before moving on to another artist - one on my current high-rotation list is Tasmanian act Enola Fall.
5. Writing
Sometimes it's good just to blow off some steam - as screaming in my office would probably cause some distress in my nearby colleagues, writing things down is a little safer. Chatting to friends online, writing blog posts, writing out to-do lists and plans - it all helps!
Friday, July 12, 2013
Mathematically possible - GWS making the AFL finals
(Update: thanks to Gazza White and the AFL subreddit for linking my post - it's already by far my most popular blog post!)
Towards the end of a sporting season, it's not unusual to hear the commentators call a team a "mathematical" chance to achieve some target - be that winning a premiership, making the finals, avoiding relegation, whatever. What this means is that there is at least one combination of events (usually discounting other teams being disqualified) that could bring it about, but it's almost vanishingly unlikely to occur.
Very seldom is this a more appropriate term than for the current chances of Greater Western Sydney getting into the top 8 and making the AFL finals this year - so much so that commentators probably aren't even aware that it is a mathematical possibility.
Here is the current AFL ladder as of the end of Round 15 (courtesy of FanFooty - note that the official AFL ladder is not actually up to date!)
In green is our team of interest - Greater Western Sydney. They are currently winless at the bottom of the ladder, 8 wins behind the lowest top 8 side (Port Adelaide - in red). Unfortunately for GWS, there are also 8 games left in the season, so one thing is immediately clear: GWS must win all 8 of their games, and Port Adelaide lose all 8 of theirs, for GWS to be any chance of making the finals (the two teams do not play each other, so this accounts for 16 separate games). If this happens, the ladder looks like this:
This on its own is still not enough to guarantee GWS a place, however - there are 9 other teams on the ladder that are also striving for a spot in the top 8. For GWS to make the finals, none of these sides can finish with more than 32 points (8 wins) at the end of the season. Therefore every game that involves one of these sides - 46 games, excluding the 16 already accounted for by GWS and Port's games - can make or break GWS's finals chances. In particular, West Coast, Carlton and Adelaide cannot get any more than 1 win for the rest of their remaining games. In fact, there are only 10 games that don't affect GWS's chances - the games between top 7 sides, who already have more wins than GWS can possibly get and are guaranteed to place above them on the ladder.
Using a computer to calculate the possible combinations in which this could happen comes up with 150,744 ways for GWS to place equal 8th. Even assuming that all teams will have a 50-50 chance of winning each game for the rest of the season (discounting draws), an assumption which is very kind to GWS to say the least, this would give them a 150,744 / 262 = 3.27 in a hundred thousand billion chance of finishing equal 8th on points.
To put this into perspective, imagine a lottery where you have to pick which 6 balls out of 40 will be drawn - a 1 in 3.8 million chance. Now imagine only entering that lottery twice in your life - and winning both times. Even THAT would be twice as likely as GWS finishing equal 8th, on a good day.
Notice that I've mentioned GWS finishing equal 8th. Even this herculean feat doesn't guarantee them a place - in the very best case scenario of the 150,744, there will be 6 teams vying for 8th place on 32 points (on average in these scenarios, there will be 9.6). So GWS's best-case scenario looks like this:
To make the finals, from this point they need to gain a higher percentage than the other 5 teams. Currently, they are on 52.21%, having scored only 1003 against their opponents' 1921 points. On the other hand, their currently best-placed opposition, Port Adelaide, has a percentage of 113.7%, scoring 1317 to their opponents' 1158.
This informative site tells us that the average score in an AFL this season so far is 92.43, and the average margin for a game is 36.92. So a roughly "average" game of AFL would involve the winner with 110.89 points and the loser with 73.97. If we assume that GWS's 8 winning games follow this scoreline, as well as Port's 8 losing games, then we end up with GWS having an improved percentage of 75.22% and Port with a dented percentage, but still plenty enough for finals, of 93.33%.
So, obviously just winning is not going to be enough for GWS to leapfrog Port and its other finals rivals. Let's assume the same as above, but this time work on the assumption that GWS has somehow found a secret scoring weapon and is able to rack up ridiculous scores while keeping their opponents to an average score of 73.97. They would need to be able to score, on average, 167.78 points in order to beat Port's percentage - an average winning margin of 93.8 on their run home - and hope that none of their other rivals have had a similar late-season percentage boost themselves. I'll leave it to someone else to work out how often a team has won 8 games in a row by an average margin of at least 93.8 in AFL history.
Our conclusion: is it possible for GWS to make the finals? Mathematically, yes. Are they going to make the finals? No. But it'd be a hell of a story if they did!
Towards the end of a sporting season, it's not unusual to hear the commentators call a team a "mathematical" chance to achieve some target - be that winning a premiership, making the finals, avoiding relegation, whatever. What this means is that there is at least one combination of events (usually discounting other teams being disqualified) that could bring it about, but it's almost vanishingly unlikely to occur.
Very seldom is this a more appropriate term than for the current chances of Greater Western Sydney getting into the top 8 and making the AFL finals this year - so much so that commentators probably aren't even aware that it is a mathematical possibility.
Here is the current AFL ladder as of the end of Round 15 (courtesy of FanFooty - note that the official AFL ladder is not actually up to date!)
| Team | P | W | D | L | For | Agt | Percent. | Pts | |
| Hawthorn | 14 | 12 | 0 | 2 | 1645 | 1167 | 141 | 48 | |
| Geelong | 14 | 12 | 0 | 2 | 1556 | 1216 | 128 | 48 | |
| Essendon | 14 | 11 | 0 | 3 | 1483 | 1142 | 129.9 | 44 | |
| Sydney | 14 | 10 | 1 | 3 | 1379 | 1048 | 131.6 | 42 | |
| Fremantle | 14 | 10 | 1 | 3 | 1201 | 954 | 125.9 | 42 | |
| Richmond | 14 | 9 | 0 | 5 | 1387 | 1190 | 116.6 | 36 | |
| Collingwood | 14 | 9 | 0 | 5 | 1321 | 1225 | 107.8 | 36 | |
| Pt Adelaide | 14 | 8 | 0 | 6 | 1317 | 1158 | 113.7 | 32 | |
| West Coast | 14 | 7 | 0 | 7 | 1404 | 1277 | 109.9 | 28 | |
| North Melb. | 14 | 6 | 0 | 8 | 1435 | 1210 | 118.6 | 24 | |
| Carlton | 14 | 6 | 0 | 8 | 1331 | 1219 | 109.2 | 24 | |
| Adelaide | 14 | 6 | 0 | 8 | 1288 | 1228 | 104.9 | 24 | |
| Gold Coast | 14 | 5 | 0 | 9 | 1197 | 1341 | 89.26 | 20 | |
| Brisbane | 14 | 5 | 0 | 9 | 1133 | 1451 | 78.08 | 20 | |
| W. Bulldogs | 14 | 4 | 0 | 10 | 1102 | 1433 | 76.9 | 16 | |
| St Kilda | 14 | 3 | 0 | 11 | 1129 | 1337 | 84.44 | 12 | |
| Melbourne | 14 | 2 | 0 | 12 | 981 | 1775 | 55.26 | 8 | |
| W. Sydney | 14 | 0 | 0 | 14 | 1003 | 1921 | 52.21 | 0 | |
In green is our team of interest - Greater Western Sydney. They are currently winless at the bottom of the ladder, 8 wins behind the lowest top 8 side (Port Adelaide - in red). Unfortunately for GWS, there are also 8 games left in the season, so one thing is immediately clear: GWS must win all 8 of their games, and Port Adelaide lose all 8 of theirs, for GWS to be any chance of making the finals (the two teams do not play each other, so this accounts for 16 separate games). If this happens, the ladder looks like this:
| Team | Points |
| Hawthorn | 52 |
| Geelong | 52 |
| Fremantle | 46 |
| Essendon | 44 |
| Sydney | 42 |
| Richmond | 36 |
| Collingwood | 36 |
| Port Adelaide | 32 |
| GWS | 32 |
| West Coast | 28 |
| Carlton | 28 |
| Adelaide | 28 |
| North Melbourne | 24 |
| Gold Coast | 24 |
| Brisbane | 24 |
| Western Bulldogs | 16 |
| St Kilda | 16 |
| Melbourne | 8 |
This on its own is still not enough to guarantee GWS a place, however - there are 9 other teams on the ladder that are also striving for a spot in the top 8. For GWS to make the finals, none of these sides can finish with more than 32 points (8 wins) at the end of the season. Therefore every game that involves one of these sides - 46 games, excluding the 16 already accounted for by GWS and Port's games - can make or break GWS's finals chances. In particular, West Coast, Carlton and Adelaide cannot get any more than 1 win for the rest of their remaining games. In fact, there are only 10 games that don't affect GWS's chances - the games between top 7 sides, who already have more wins than GWS can possibly get and are guaranteed to place above them on the ladder.
Using a computer to calculate the possible combinations in which this could happen comes up with 150,744 ways for GWS to place equal 8th. Even assuming that all teams will have a 50-50 chance of winning each game for the rest of the season (discounting draws), an assumption which is very kind to GWS to say the least, this would give them a 150,744 / 262 = 3.27 in a hundred thousand billion chance of finishing equal 8th on points.
To put this into perspective, imagine a lottery where you have to pick which 6 balls out of 40 will be drawn - a 1 in 3.8 million chance. Now imagine only entering that lottery twice in your life - and winning both times. Even THAT would be twice as likely as GWS finishing equal 8th, on a good day.
Notice that I've mentioned GWS finishing equal 8th. Even this herculean feat doesn't guarantee them a place - in the very best case scenario of the 150,744, there will be 6 teams vying for 8th place on 32 points (on average in these scenarios, there will be 9.6). So GWS's best-case scenario looks like this:
| |||
| Geelong | 76 | ||
| Fremantle | 66 | ||
| Hawthorn | 64 | ||
| Essendon | 60 | ||
| Sydney | 54 | ||
| Collingwood | 52 | ||
| Richmond | 48 | ||
| Port Adelaide | 32 | ||
| Carlton | 32 | ||
| Adelaide | 32 | ||
| Gold Coast | 32 | ||
| Western Bulldogs | 32 | ||
| GWS | 32 | ||
| West Coast | 28 | ||
| North Melbourne | 28 | ||
| Brisbane | 28 | ||
| St Kilda | 28 | ||
| Melbourne | 28 |
To make the finals, from this point they need to gain a higher percentage than the other 5 teams. Currently, they are on 52.21%, having scored only 1003 against their opponents' 1921 points. On the other hand, their currently best-placed opposition, Port Adelaide, has a percentage of 113.7%, scoring 1317 to their opponents' 1158.
This informative site tells us that the average score in an AFL this season so far is 92.43, and the average margin for a game is 36.92. So a roughly "average" game of AFL would involve the winner with 110.89 points and the loser with 73.97. If we assume that GWS's 8 winning games follow this scoreline, as well as Port's 8 losing games, then we end up with GWS having an improved percentage of 75.22% and Port with a dented percentage, but still plenty enough for finals, of 93.33%.
So, obviously just winning is not going to be enough for GWS to leapfrog Port and its other finals rivals. Let's assume the same as above, but this time work on the assumption that GWS has somehow found a secret scoring weapon and is able to rack up ridiculous scores while keeping their opponents to an average score of 73.97. They would need to be able to score, on average, 167.78 points in order to beat Port's percentage - an average winning margin of 93.8 on their run home - and hope that none of their other rivals have had a similar late-season percentage boost themselves. I'll leave it to someone else to work out how often a team has won 8 games in a row by an average margin of at least 93.8 in AFL history.
Our conclusion: is it possible for GWS to make the finals? Mathematically, yes. Are they going to make the finals? No. But it'd be a hell of a story if they did!
Monday, July 8, 2013
Music
After the joy of launching an EP (look to the right of screen - that's my EP), my rock music career has quietened down substantially in the last few months. Initially, I wanted to concentrate on writing songs suited more to my band (The Solution), but the band has itself faded into the background a little after our bassist moved to the other end of the state for work. We're still getting the occasional practice session in, and are steadily working towards recording an album, but it's left a lot of time in which to ponder other musical directions.
One of these has been the choir I joined last year - the Tasmanian Song Company. When I joined, I sang in the tenor section but as the number of males in the group has grown (due in part to some of my friends joining!), it became obvious that we needed more basses so I moved there instead. As time's gone on, I've found my involvement growing to the point where I found myself joining the committee and helping out on a regular basis. I've never been on any kind of committee before, but this one involves cake and cups of tea so it can't be all bad!
The other way I'm keeping myself going with music is busking. It had been a long time since I busked, so a month ago I put together a collection of covers and made my way out to Elizabeth Mall - and I've been trying to get out there every week or so. It's a great way to practice performing in front of people - something I sorely needed when I was a beginning musician years ago, but just as useful now that I've got a little more experience and want to keep my skills under pressure fresh.
Though I'm fortunately not broke enough to need the money from busking, I still find it a good way to "keep score" of how well I'm going - of course, it doesn't hurt if I make enough to buy lunch and have some change for parking meters! Over the weeks, though, I've found the money really isn't a good measure of how people are reacting to my music. A couple of weeks ago, I went out on a crowded day and only made a couple of dollars despite singing my heart out, and I was feeling pretty miserable about the whole affair. Then, in the middle of my set, an obviously down-and-out, slightly elderly lady came up to me and said very sincerely "Lovely singing - I'm sorry I don't have any money to give you."
Since then, I've gotten far more joy out of playing music I love out in the winter sun, getting a smile of recognition or a kind word from a passer-by, or watching small children dance gleefully in front of my guitar case. Sometimes it doesn't hurt to be reminded of the old cliché that money doesn't buy happiness!
One of these has been the choir I joined last year - the Tasmanian Song Company. When I joined, I sang in the tenor section but as the number of males in the group has grown (due in part to some of my friends joining!), it became obvious that we needed more basses so I moved there instead. As time's gone on, I've found my involvement growing to the point where I found myself joining the committee and helping out on a regular basis. I've never been on any kind of committee before, but this one involves cake and cups of tea so it can't be all bad!
The other way I'm keeping myself going with music is busking. It had been a long time since I busked, so a month ago I put together a collection of covers and made my way out to Elizabeth Mall - and I've been trying to get out there every week or so. It's a great way to practice performing in front of people - something I sorely needed when I was a beginning musician years ago, but just as useful now that I've got a little more experience and want to keep my skills under pressure fresh.
Though I'm fortunately not broke enough to need the money from busking, I still find it a good way to "keep score" of how well I'm going - of course, it doesn't hurt if I make enough to buy lunch and have some change for parking meters! Over the weeks, though, I've found the money really isn't a good measure of how people are reacting to my music. A couple of weeks ago, I went out on a crowded day and only made a couple of dollars despite singing my heart out, and I was feeling pretty miserable about the whole affair. Then, in the middle of my set, an obviously down-and-out, slightly elderly lady came up to me and said very sincerely "Lovely singing - I'm sorry I don't have any money to give you."
Since then, I've gotten far more joy out of playing music I love out in the winter sun, getting a smile of recognition or a kind word from a passer-by, or watching small children dance gleefully in front of my guitar case. Sometimes it doesn't hurt to be reminded of the old cliché that money doesn't buy happiness!
Subscribe to:
Posts (Atom)