A short interlude from all the recent maths…
When I was at university I tread the boards regularly, usually two or three times a year and I never slept with the director - Which is probably why I was always restricted to limited roles with the exception of the time I was assisting the director and the play found itself without without a lead male and necessity led to me being promoted from dogsbody to leading man.
I haven’t been on stage for a while though and am unlikely to be in the future. I am under no illusions as to my acting ability or rather lack of acting ability. I remember one play I was part of which was set in the Westcountry, the only person in the cast who couldn’t do a Westcountry accent was me and I was of course the only member of the cast from the Westcountry.
Indeed the only comment that I can recall being made about my acting skills is, “You come from the Brian Blessed school of acting.” A comment that to this day I cannot decide was meant as praise or criticism.
Still I have been given hope recently that I may one day make it back onto stage and fufill an ambition of mine, by performing Shakespeare on stage. To be fair performing Shakespeare is perhaps stretching the term a fair bit as the role I would be ‘performing’ would be that of Yorick.
All I have to do is follow in the footsteps of André Tchaikowsky and bequeath my skull to a theatre company and wait 26 years for them to pluck up the courage to use it.
This is part 2 of a 3 part ramble of mine now that it’s complete here are the links to each part to make it easier to see what it links to.
Part 1 - The Schrödinger Cat Game
Part 2 - You’re reading it now.
Part 3 - Sticking or Swapping?
Previously I discussed some of the mechanics of Deal or No Deal and how it relates to Schrödinger’s Cat as an exercise in indeterminism, as a quick refresher here are a couple of examples for you to think about what you’d do if paying deal or no deal.
- The are 2 boxes in play, £100 and £10,000. The banker offers £3,800. Do you deal?
- There are five boxes still in play, the amounts left are £10, £100, £1000, £10,000, £100,000. The banker has offered you £12,500. Do you deal?
In the first scenario, you have the choice between taking the money or effectively gambling for all or nothing after £100 is only 1% of what you could win potentially win. The bankers offer is less than half the statistical winnings of £5,050. So statistically speaking, gambling is the right choice as over time you would win more – Of course you are not gambling over time and only have the one chance, so you could lose it all. From a more personal viewpoint, the amounts are unlikely to be radically life changing and you aren’t going to lose them. It’s just that you might not win them, which again suggests that going for the gamble is the best option for the contestant and it’s not merely being greedy.
The second case is even less clear cut, even though the Banker’s offer is £10,000 below the statistical winnings which at this point in the game would be £22,222. At first glance I thought I would deal with almost no hesitation. Firstly the statistical winnings only take account of one average, the mean.
Whilst useful as a starting point it can be skewed by excessively large or small values and here the only reason the statistical winning is £22,222 is because one of the prizes is over 9 times greater than the sum of the other four prizes put together. The bankers offer seems low compared to the statistical winnings, but note that it is still higher than 4 out of the 5 potential prizes. This means that if you didn’t deal and played to completion (ignoring for the moment that you would get another offer when down to two boxes) that you have an 80% probability of walking away with less and so cutting your losses at this point seems like a sensible course of action. You might only win £12,500, but if you play on it is a lot more likely that you will win less.
However can the odds be improved? Technically answer is no, but if you rephrase the question it continuing to play doesn’t seem like such a bad idea. In the example the bankers offer is reasonable close to one of the prizes and if one treats them to be the same the question changes slightly. Think about it, what is the difference between £10,000 and £12,500 in real terms to you? A few more shiny things perhaps, but the difference is negligible – Or at least it is for mathematicians who are used to treating horses as infinitely small points traveling at fixed velocities. Then the odds become:
- 20% Win more
- 20% Win the same
- 60% Win less
So playing to completion as it stand means the odds are only slightly worse than even that you are going to worsen your situation significantly, but that you have a 1 in 5 chance of winning big.
It’s still more likely that you will win less overall, but this decision is weighted to take into account the values in relation to each other and so isn’t quite as all or nothing and if playing to completion only I’d take those odds and gamble on my box.
But that is no how the game is played, we know that the banker will make another offer when we get down to just the two boxes and after that we have a chance to swap boxes, so it’s not just a case of playing to completion or dropping out now. At this point of the game however it’s still in the future and although indeterminate we can look at the probabilities and see if it’s worth going further.
Now although one box is fixed and you can’t open it to find out the contents, you can open three of the other boxes to remove possibilities. Taking advantage of the undefined nature of the box contents we can treat all contents as random. After 3 of the boxes are opened we are left with 20 possible combinations for contents of the boxes, despite there only being 4 possible combinations of boxes.
Allow me to demonstrate, I have box number 12 and the boxes in play are numbered 3. 8. 19, 20. (The numbers on the boxes don’t matter, I am just using them as examples.) After revealing three of the boxes I am left with the following 4 combinations:
I would also know the potential contents of the 2 boxes and then be able to work out how best to proceed. Realistically at this point if you have been lucky and the £100,000 is still in play you should get a banker offer of between £35,000 and £55,000 and if it was me I would take the money and run at that point and not risk the gamble. However if I had ridden my luck too hard and removed the £100,000 from the equation I might as well gamble on the contents of a box and hope for the top remaining prize as the amounts don’t really matter too much at that point. Unless it a choice between £10 and £10,000 for example with a banker offer of £5,000 – Although as at this point the game is considered lost I will not model the decisions further.
However with the five prizes still in play, you don’t know the exact situation you will find yourself in once you have got rid of 3 of the remaining boxes. Assigning the letters A, B, C, D and E to the contents of the boxes, the possible outcomes are:
AB BA CA DA EA
AC BC CB DB EB
AD BD CD DC EC
AE BE CE DE ED
However as the contents of the boxes are not known, the order is not important. AB is the same as BA. After all £100,000 in a box and £10 in the another box is the same as £10 in one box and £100,000 – Until you open one of the boxes they both contain £10 and £100,000, the cat is both alive and dead until you look at it. What this means is that the 20 combinations become 10 as they pair up with each other, this leaves:
AB BC CD DE
AC BD CE
Now if we assign a letter to each prize so that A=£10, B=£100, C=£1000, D=£10,000 and E=£100,00, we can see that the chance of leaving the £100,000 in play (and indeed any specific prize) is 40%. The chance of leaving the £10,000 in play is also 40%, however as there is a chance that both the £100,000 and £10,000 are left in play the chance of £10,000 representing winnings is actually 30%. The remaining 10% represents the £10,000 being ‘losings’.
So at the moment we have:
- 40% chance that top prize is £100,000.
- 30% chance that top prize is £10,000.
- 30 % chance you’ve flogged the horse of chance to death.
Assuming that you haven’t ridden your luck too far and are still in the game, what this boils down to is:
- 40% chance the banker offers approximately £50,000
- 30% chance the banker offers approximately £5,000
- 30% chance the banker offers a negligible amount
The statistical winnings using this model are (0.4 x 50000) + (0.3 x 5000) + 0 or as a numeric value, £21,500. This number is still heavily biased due to the large value of one prize compared to the others, but it allows you to weigh up the odds.
So contestant you have the choice walk away with £12,500 or gamble to the next stage potentially losing most of it. Obviously one part of the decision relies upon how important £12,500 is to you. Whilst probably not a life changing amount, if used wisely it could alter ones lifestyle over time or give a chance to have some experiences that otherwise might remained unfulfilled or in these days of credit meltdown might offer a fresh start. If the last reason is the case you’d be a fool to gamble further, for me however the choice is between buying shiny things or using it judiciously – Either way things I currently get by without.
In short I’d gamble as I have a 40% chance of increasing my winnings by a factor of 4 and those odds are good, especially with a potential fall-back position where I have a chance of still walking away with the second place prize of £5,000 if things don’t pan out and only a 30% chance of effectively winning nothing.
In fact with Deal or No Deal, statistically speaking you should lamost always gamble, over time you would win more than the banker will ever make as an offer, as the offer should always be less than the average (mean) winnings – As with gambling, the house always wins. However using the median is also worth contemplating especially if one of the prizes is disproportionately small or large and if all prizes except one are less than the banker’s offer you probably won’t be walking away with the biggest prize and might want to consider not being greedy and taking the guaranteed amount unless you wind up with a case like the one above which does offer a fallback position of sorts.
However I still have got to the question that started this whole train of thought off – Should you swap your box at the end of the game? I’ll get to that next time.
Originally this post was supposed to be an explanation of something called the Monty Hall problem and how people aren’t very good with probabilities, which I was explaining to a friend at 6 in the morning. It has turned into a monster. However, I have sidetracked myself discussing other aspects of the problem and after 2,000+ words into my explanation I have not touched upon my original topic. On the plus side, I have proved some things that I suspected and so now actually know them to be true. I’ll put the rest of this essay up in manageable pieces rather than put it all up at once.
If after reading this part you want to read more here are the links to parts 2 and 3 of this ramble.
Part 2 - To Deal or Not To Deal?
Part 3 - Sticking or Swapping?
The other morning I wound up discussing Mathematics, or more precisely I was engaged in conversation about probability at 6am. Now there is never a right time of day to talk about statistics and probability as far as I am concerned, but this goes double when you’ve been up all night beforehand.*
The conversation started when my friend make a comment for reasons I cannot remember about the ‘Schrödinger Cat Game’, or Deal or No Deal as most people would know it. The analogy is actually quite good to be fair, as both involve randomness and indeterminate states.
For the uninitiated, Schrödinger’s cat is the subject of a thought experiment, a cat that can be both dead and alive at the same time. The idea is such that you put a cat in a sealed box. The interior of the box is unobservable from outside, but has everything a cat needs to survive – The catch is the food, there are two sources of food that can be given to the cat, one of which is poisoned. Which the cat is given is decided by chance and it’s 50/50 which food the cat will be given.
The process of determining which bowl of food is given to the cat is random and decided by the half-life decay of a radioactive substance, something which is completely random. So after given a bowl of food the cat is either dead or alive and quantum mechanics suggests that the cat is simultaneously dead and alive. Yet when the box is opened the cat is either dead or alive, as although quantum physics may permit something to be both dead and alive, biology doesn’t.
Deal or No Deal is somewhat akin to this, in that the 25 boxes each contain 25 differing amounts and up until the point of opening, the contents could be any one of the amounts. Now as it is a real life scenario someone has to place the amounts in the boxes and the states are known, but for the purposes of this model I shall assume that the contents of the boxes is determined randomly and nobody is privy to the contents.
During the course of the game the 25 boxes are whittled down to just 2 and we are effectively down to the same scenario as with Schrödinger’s cat. We have two boxes with differing contents, however under quantum mechanics, until one box is opened and it’s contents revealed both boxes contain both prizes. The revealing of the other 23 boxes is irrelevant, except to tell you what states are not possible for the last two boxes. Of course the possible states are important, after all you are not choosing between two unknowns you also have the guaranteed bankers offer as well.
Look at the following two idealised examples to see what I mean.
One box has 1p in it and the other has £250,000. The bankers offer is for £125,000.**
One box has £10,000 and the other has £20,000. The bankers offer is for £15,000.**
In the above examples, polarised as there are three solutions; keep the box you had at the start, swap the box for the other box whose contents you also don’t know or take the bankers offer. In the first example the solution for anyone who isn’t foolish or greedy is to take the money offered by the banker. Even in a non idealised example where the offer might only be £100,000 it still makes sense to take the money offered as it is guaranteed, after all if you don’t take it you could walk away with nothing.
The second case is a little less cut and dried and depends upon the individual after all not as much is riding on it after all in this case the worst result still sees you £10,000 richer and whilst the reward is less, the risk is less and you can afford to be greedy and gamble on the potential contents of the box.
Assuming that the person gambles there is a final question, should you swap your box? Or rather does it make any difference if you swap your box?
As I’ve gone on for quite a bit already, I’ll leave that for you to think about for the moment and put my answer and explanation for it later.
*I make be a bit of a maths freak, but I wouldn’t get up at 5am just so I could discuss it with someone.
**These are idealised offers based on the statistical winnings over time, the bank never offers these as the house always wins. Usually the bankers offer would be 10-20% lower than the statistical winnings. After all guaranteed money is better than the chance of getting more money (or it is if you take it anyway). Remember a bird in the hand is worth two in the bush.
Because I know if I don’t say something about this article, I’ll probably get sent links to this a few times over the next couple of days - For the record O would just like to state that this is not the reason for my fascination with Japan.
Bra for the boys an online bestseller in Japan
TOKYO (Reuters) – Who said bras are only for women? A Japanese online lingerie retailer is selling bras for cross-dressing men and they’ve quickly become one of its most popular items.
Since launching two weeks ago on Rakuten, a major Japanese web shopping mall, the Wishroom shop has sold over 300 men’s bras for 2,800 yen ($30) each. The shop also stocks men’s panties, as well as lingerie for women.
“I like this tight feeling. It feels good,” Wishroom representative Masayuki Tsuchiya told Reuters as he modeled the bra, which can be worn discreetly under men’s clothing.
Wishroom Executive Director Akiko Okunomiya said she was surprised at the number of men who were looking for their inner woman.
“I think more and more men are becoming interested in bras. Since we launched the men’s bra, we’ve been getting feedback from customers saying ‘wow, we’d been waiting for this for such a long time’,” she said.
But the bra, available in black, pink and white, is not an easy sell for all men.
The underwear has stirred a heated debate online with more than 8,000 people debating the merits of men wearing bras in one night on Mixi, Japan’s top social network website.
From Yahoo news.