Heads or tails? It’s a simple question with a far from simple answer. One that takes us into the strange and complex world of probability.
Probability is the field of maths relating to random events and, although commonplace now, the idea that you can pluck a piece of maths from the tumbling of dice, the shuffling of cards or the odds in the local lottery is a relatively recent and powerful one. It may start with the toss of a coin but probability reaches into every area of the modern world, from the analysis of society to the decay of an atom.
Contributors
Marcus du Sautoy, Professor of Mathematics at the University of Oxford
Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews
Ian Stewart, Professor of Mathematics at the University of Warwick
Audience reactions to this edition
Dr Jonathan Ovens
As a fan of IOT I have struggled a little to find the right words to say in response to last weeks program on Probability. I think the program tried to cover too much ground and in so doing missed several very relevant ideas and got sidetracked onto other ideas which, consequently, were delt with in a very shallow perfunctory manner. We went from a detailed discussion of Birth Dates to skating over the EPR paradox and chaos theory all in 40 minutes. It would have been much more satisfactory if we had stayed with the likes of Bayes and explored things like conditional probability.Furthermore, to the extent that 'probability' lies within the branch of applied maths known as statistics and statistical descriptions of processes range from geology to Wall Street, the program really only touched the tip of an enormous iceberg.
Mike Disney, Probability is far more than mathemat
Your program, while interesting as always, so thank you, suffered from the grave handicap of having only mathematicians talking about it. The subject has moved far on beyond mathematicians with their comfortable little problems of dice and cards and birthdays.Science, for instance, needs to draw inferences from incomplete and imperfect information, and to do that one needs to redefine 'Probability' in a much broader sense than mathematicians will generally allow, or are comfortable with. One such definition is 'The degree of rational assent one can give to..." (some proposition).Thus Laplace (1812) had to completely, if coyly, redefine the meaning of Probability away from the narrow mathematical sense of your program, to adapt it to Science, and indeed to much of everyday thinking. For instance he and Bayes showed that you need to incorporate "Priors" i.e. "Prejudices" into your Probabilistic Thinking [otherwise known as 'Common Sense'] in order to reach conclusions. Mathematicians have generally hated that idea, and tried to deny it, and this has brought about a huge schism within the discipline of 'Statistics', much of which must be wrong. Though which bits you believe are wrong will then depend on the School of Probability to which you adhere: 'Frequentist', 'Objective Bayesian', Subjective Bayesian', and so on. Probability is, I find as an astrophysicist, a far harder concept to deal with than either Relativity or Quantum Physics, and is arguably far more important than either for it underlies nearly all of our thinking, scientific or otherwise.
Oli- Re: Heidi P & the Birthday Problem
The way the panel suggested was best to work this out was to look at the reverse problem- how many people can you have in a room before the probability is <0.5 they will all have different birthdays.(ignoring leap years etc) The suggested method took the probability of 1 person (=1)2 people (1 * 1/365)3 people (1 * 1/365 * 1/364)4 people (1 * 1/365 * 1/364 * 1/363)etc.From memory, the number of people needed to be in the room before the probablilty of any 2 sharing a birthday was more like 24. A surprising and counter-intuitive result.
Oli- Why the focus on gambling?
I would loved to have heard more on modern and topical uses of probability- perhaps the early beginnings of medical research and why such trial data is stronger evidence than anecdotal evidence?There is more to statistics & probability than just gambling...
Charles Collingwood - "Three Doors" problem rethin
Thanks to Dave Howe and Tom Milner-Gulland I had second thoughts about this subject, and realised that my ealier confidence was quite wrong, and that switching is indeed the right answer.Furthermore, the reference to Monty Hall led me to look it up on the internet, and I am now a bit wiser about probability problems, and certainly a lot more cautious. Thanks to 'In Our Time'.
Liam in Spain - The Odds of this weeks show
As I am currently studing probability and insurance calculcation for my thesis, and as I know that "In our time" touches many topics of philosopical and scientific interest, what indeed are the odds that the most recent show would concern probability? I had not thought about that question until I saw today that I did not need to dip into the archive to find a program on my subject because the most recent show nicely dealed with my current area of study!
David Tranah, In our Time on Probability and Stati
I was excited that these difficult but extremely contemporary and important subjects were to be tackled. All of science and much of social science is going through a probabilistic revolution. Why not then get some experts to talk about them?
David in Brussels Probability in his
Why is probability only a comparatively modern study? By averaging and cutting data to a minimum, probability provides a useful mathematical razor for physics, gambling and sociology. Our enfeebled perceptive abilities take in only a tiny fraction of all data (we don’t see X-rays or infrared). We then select and simplify a small fraction of this perceived data for probability calculations. Approximation, and hence probability, cannot reflect total reality or truth. Quantum physics based on probability showed something apparently contradictory to approximation. Nuclear particles, when split, retain a quantum relationship when they travel far apart. Experimental evidence confirmed that information travelling faster than light interconnects all matter. This explains why radioactive material has a probabilistic half-life; “random” decaying particles all interact with themselves. The ensemble contributes to regulating an atomic clock. Modern physics thus supports interconnectivity of matter, a dominant idea among non-atomistic (or non-Greek) philosophers of science throughout the millennia. In recent times Soviet Communists to attack Christianity and the open society gave ideological support for a philosophy based on the probability of quantum physics. That was dishonest. What works more or less in physics based on a razor fallacy is insufficient for universal philosophy. Randomness has yet to be proved to exist. The mathematics of wise, universal interconnectivity is completely beyond our feeble brain or the best computers (both of which are also part of the variables). This is why we select. If randomness does not exist, do we have free will? As a graspable example of a universal intelligence, one can turn to a series of probabilities recorded in the Jewish Babylonian Talmud yoma. In the last forty years before the destruction of the Jerusalem Temple in 70 CE and the scattering of Israel, the lot designating the sacrifice for the nation on the Day of Atonement always turned up at the high priest’s left hand. It was a consistent ill omen, a warning for repentance, as around the time of the righteous high priest, Simon, the lot had previously fallen on the right for a similar forty-year period. The probability for such long series is extremely small; it amounts to considerably less than one in the total number of humans who have ever lived. Probability (as razored, ill-informed expectation) could have been considered each yearly event as evens (left or right). A longer series, also dependent on movement, exists. In each of its 1878 years of exile, a bookie or statistician would have considered the prophesied re-gathering of Israel as a State was even more improbable than evens.
Jane - probability
Haven't had time to read all this week's comments so I may be repeating something but just as the scientist is said to '"affect the experiment" do we affect the randomness? I don't mean by the way we throw the coin etc.. Also, semantics pin us down here - the word 'randomness' is perhaps too narrow by which I mean that we are isolating it into a phenomenon when in fact, did we have more insight, it would be a case of phenomena. best wishes
Tom Milner-Gulland - 'door' problem
A way that the Monty 'door' problem is commonly explained is that if you have a million doors and Monty opens 999,998, excluding that which you yourself have chosen, and one other, then you would be foolish not to switch doors, the rules -- these being that one of the the two doors *must* have the car behind it -- indicating the overwhelming odds in favour of switching. I thought of an interesting variation upon this. Consider that the 999,998 doors were opened *at random*. This would mean it is pure chance that those particular doors -- yours and the one other --remain closed. Now, if you played this game many times, the random-opening process will leave closed the door behind which is the car twice as often as you select it. The first door it randomly selects to be closed has a chance of one in a million, as does the second, meaning the odds are even between each, irrespective of whether you yourself have selected it. Therefore, in the familiar three-door problem, once one door has been opened, if you then know whether or not the door-opening process was at random, you know whether or not it is better to switch doors. That seems a strange conclusion, as one might think that once the door has opened it has opened; the determinant would seem irrelevant. In fact, the process of my selecting, here, only those occasions in which the door that opens is a door that Monty might have selected, is what makes the difference in terms of odds.
Martyn Wilson, Probability
In Melvyn's newsletter, he says that "the sequence of numbers: ‘1, 2, 3, 4, 5, 6’ [in the National Lottery] are just as likely to come up as any other combination of numbers". How? The first number is as likely to be one of these six as one of the other 43. However, thereafter, the choice of numbers reduces, and the odds therefore increase exponentially. If the first number is (say) 3, the next number is not randomly selected from one of the 48 remaining numbers, but randomly selected from 1, 2, 4, 5 and 6. If it happens to be (say) 6, then the third number must be selected from 1,2,4 and 5. And so on.It actually conflicts with what Melvyn goes on to say, "that numbers are more likely to come up in clusters". 1, 2, 3, 4, 5 and 6 is the ultimate cluster. Incidentally, on the basis of empyrical evidence, this does not make sense either: in the past 20 draws for the National Lottery, two adjacent numbers have occurred four times, and three adjacent numbers not at all.
chris; newsletter on probability
Melvyn's Newsletter, always interesting and helpful,occasionally invites nitpicking, e.g. "randomness likes to form in clusters". How thoughtful of it. Perhaps a future subject could be: the tendency of scientific enquiry to use anthropomorphism. This particular sentence also contains a circular argument but let that pass.
Iain Tokyo, improbability
all this talk of coin tossing reminds me of a particularly bizzare event I witnessed many year ago. While on a long distance train a traveling companion dropped a handful of coins, one of which landed on its edge. The probability of a dropped coin landing on its edge is beyond my ability to calculate. On a moving train seems impossible. It seems that no matter how improbable, anything not logicaly excluded, is posssible and can happen. Such improbability is however a much more difficult thing to quantify that probability. It may be a much more interesting issue.
Heidi P: Birthday Problem
I'm sure Ian Stewart said that the number of people needed to be in a room with me before the odds of someone else there sharing my birthday is >0.5 was 250. Is this correct? Surely the chance of the first person also having the same birthday (last wednesday, actually) is 1/365. And of the 2nd, also 1/365 and so on. To get the total probability you need to add these up, since they are independent events. So the probability of someone sharing my birthday when there are n people in the room is n/365. And 183 people are needed for the probability to be >0.5. Am I missing something?
Penelope /Probability
Give me mathematicians any day! No hesitation, no waffling, no ums and ers, but clear, concise, crisp, illuminating,, entertaining and brilliant. Thank you.
Steve B - Roulette; Zero is not the House's number
Melvyn made an often made but erroneous comment about Zero being the House's number in roulette, implying that whenever the ball lands on zero then the Casino was at its happiest because it made most money out of the punters. This isn't true, particularly in relation to someone who is betting on individual numbers as opposed to Odds/Evens, Red/Black, 1/3 bets and I'm surprised that he wasn't challenged on this by the panel. The correct explanation was actually given in the general discussion that took place - it is from the punters' winning bets that the casino makes its money on by only paying out odds of 35 to 1 when there are 37 possibilities on the table (and so should be paying 36 to 1 for a completely fair return). Of course they wouldn't do this because over the long term they would only break even and therefore go out of business. The issues surrounding the Sally Clark story really are ripe for further discussion. Probabilities really can be so misleading when either i) are not understood or ii) are based upon assumptions about inappropriate reference sets. A programme examining the presentation of statistics and probabilities compared with using natural frequencies (see "Reckoning with Risk" by Gerd Gigerenzer) would make fascinating listening.
Mark Tillotson, Probability
Good program, but I have to comment on Melvyn's newsletter which seems to fall into the kind of counter-intuitive trap that the show highlighted rather well!Talking of choosing lottery numbers, Melvyn rightly points out that choosing numbers that other people might choose reduces your chance of winning a big prize (greater risk of having to share it)However going with consecutive numbers is a bad example - in general lottery entrants are likely to pick sets of numbers with more structure (less entropy), so to increase your chance of a high-value win you need to pick a high-entropy set of numbers (at random, in fact).Of course your chance of winning isn't affected by any structure in your set of numbers, just the chance you might have to share the pot!
Chris Myant probability/lysenkoism
Melvyn in his news letter uses the 'if a butterfly flaps its wings in the Amazon there can be a hurricane in Tokyo' idiom which is a poor expression of the point as hundreds of millions of butterflies flutter around all over the world and there are only a few hurricanes in Tokyo. A better illustration is the natural selection point which Stalin could not accept - because he could not control it - that a small accidental change in the DNA of one item of life at time x will affect a growing number of living things at time x to the power infinity. We look at it now, looking backwards in time, and it looks ordered and not chaotic because the consequences are built in to the life around us. The randomness of the initial change has been countered by the degree to which it copes with the other factors in the context in which it finds itself. If we look forward on the other hand, we appear to see only chaos in which a random change has unpredictable but potentially apparently unrestricted results. Stalin wanted to be able to reshape nature to fit his idea that the world was completely maleable if only we souihg to make it so. Lysenko's ideas fitted in with that apprently 'optimistic' view of the world. The alternative appeared to mean that we needed to hunt through the jungles of the Amazon to find the one butterfly whose one wing flap would set in motion the causal chain leading to the hurricane in Tokyo, an obviously impossible task.
Dave Howe 3 doors problem
I notice some people seem to have misunderstood this (some emininent mathematicians have fallen into this trap). When the contestant decides to switch, they don't take another random guess from the 2 remaining doors. This is a different problem and would give a 1/2 probability of winning (but note this is still better than the 1/3 prob. of the first choice winning). If you've tried this on a computer and found a 1/2 prob. the computer must have been given a different problem to solve.Melvyn rather rushed things on at this point in the programme. Put simply: If you stick there's 1/3 prob of winning. Once a goat is revealed, the only other place the car could be is behind the other unopened door. The car must be behind one of the doors, so there's a 2/3 prob. it is behind the other unopened door - so you should switch.
Colin Milne - Probability.
Dear Sir, Re: Probability. """""""""""""""" There is a difference between random and unpredictable. " Random " is by definition always unpredictable: " unpredictable " is, possibly, predictable with more information. The " Heads or Tails " problem is interesting, because the 1/2 probability must pertain throughout the " flip ". Really, it is just a measure of imperfection and functionality. The electronic oscillators are still functional, and it is possible to adjust them over periods to show that they are 50/50.The errors are because of the imperfections. A coin should, therefore, alternate each time!!!!It is a philosophical matter ! Yours faithfully, Colin P. Milne.Location - Bebington Library.
Tom Milner-Gulland - Probability
The issue of probability is made especially counter-intuitive by virtue of our human tendency to identify an event first, and ascribe a probability to its occurrence subsequently. But suppose we look at it the other way: we, as humans, identify, conceptualise and apply prescriptive terms to types of event - such as the weather, the tossing of a coin, a sporting event - not by virtue of the nature of their content, but by virtue of that content's exhibiting a discernible probability distribution. Empirical outcomes follow a pattern that only *appears* to contain a random component, because we are naturally disposed to discern and isolate phenomena, wherein we might identify probability distributions, and apply an internal system of logic, as though the phenomenon were self-contained and not a mere part of a holistic cosmic system. True randomness is not found in reality; it would elude any determinable probability distribution. --- BTW in reply to an earlier commentator, the Monty Hall 'door' problem was correctly analysed; it all hinges (PI!) on the fact that the determination of which door is first opened is not random.
Diana Gillooly - probability + book recommendation
An excellent choice of subject, but I agree with Les Weston that too much was attempted in one program. I hope that IOT will follow up some of the rich possibilities. If/when they do, I very much hope that at least one of the guests will be an expert in the field. Karen Radner was brilliant in the Nineveh program; a working probabilist or historian of probability could bring similar depth and verve - e.g. Lorraine Daston, who wrote a great and entertaining book: Classical Probability in the Enlightenment. One point it makes is that probability is a vehicle for comparing completely unlike things - read about the bet between two earls: that one could ride from London to Edinburgh and back before the other could make a million dots however expeditiously.
probability - coin tossing
a skilled "tosser" can judge the degree of flick given to a coin - knowing this a coin can be made to fall heads only or tails only or any desired combination......therefore a tossed coin is not a good example to use for examples of randomness
Steve H; Bahman Farzade's question
I'm not aware of these but this comes down to the argument about when something is truely random. Taking the weather as an example, the weather on a set day and time 100 years hence will look random to us from our standpoint of today, but the weather 5 seconds from now will not, because we have sufficient information to predict it. In tossing a coin we are unable to predict how many times it will turn in the air before it lands. However, I could probably design a machine that tosses a coin with a very precise (i.e. reproducible) mechanism that would allow me to get the same number of turns each time and for the coin to fall in a set way. This becomes increasingly difficult with an increasing number of turns, because the butterfly effect is more prominent, but I could probably manage it for 3 or 4 turns and a low height of tossing! Does this help?
Steve H; Probability-Robert Fallon's comments
If you have not tossed your coin yet, the chance of a head is 1/2. If you've already thrown your coin 6 times, regardless of the results, the chance is still 1/2. If you tossed the coin 6 times and got 6 heads (a chance of 1 in 2^6) you still have a 1/2 chance with the last throw. Each possible sequence of results from 7 tosses has exactly the same probability of occuring (i.e. 1 in 2^7) but by the last toss you have only 2 possible outcomes (of equal probability) because the other 6 have been determined already. Probability is the most amazing thing to study and is the one of the reasons that being a scientist is endlessly fascinating. Probability/statistics is commonly counterintuitive, but that just makes it all the more amazing when the light go on!
Steve H; Proability and the Lottery
Melvyn's newsletter suggested that selecting clustered numbers improves your chances of winning the lottery. This isn't true. The winning numbers are more likely to be clustered because there are more clustered then 'well spaced' possibilities.
Richard Simpson: Probability
A fascinating programme that could be made into a series. There was no mention of Pareto's 80-20 law and I wondered if this has been debunked. I would be surprised if it has because it seems to be used a lot and it does tick the 'counter-intuitive' box of probability.
Dr Ken Sullivan - Randomness is an illusion
Is there a recognised hypothesis that states that 'randomness' does not actually exist? Isn't randomness actually just an illusion - a consequence of chaotic, but nonetheless deterministic, systems?Quantum effects (such as radioactive decay) would also be an illusion within this hypothesis. If there was indeed a 'random' mechanism in nature, then what would be the nature or the origin of this machanism? I.e. how could randomness come about, without a superior intelligent agent? There is a strong parallel here with the 'free-will is an illusion' argument. In fact, one could claim it to be the same argument.
Alex Iden - Probability.
Toss a coin several times in succession. You get three heads in a row and are now about to toss again. Do you feel, ‘in your bones’, that some sort of ‘pressure’ has now accumulated that will encourage the coin to come down tails this time? The pressure of ‘destiny’ perhaps? How else can the 50/50 balance be redressed?What does this tell us about how our minds work, and the constant interplay between emotion and logic?
Bruce Mardle; definition, quantum physics and inde
I enjoyed the program but I think it would have been better if it included a definition of 'probability'. I'm familiar with 2 meaningful definitions (Frequentist and Bayesian) but they don't seem to cover what people often mean by the word. Most of the definitions I've seen are in terms of 'odds', 'chance', 'likelihood'... which is just passing the buck.As I understand it (OU physics course) most physicists believe that nature is either 'local' (i.e. no signal can travel faster than light) or 'real' (things have properties before they're measured) but not both.As to the probability of sequences of heads from an ideal coin, *having thrown 5 heads*, the probability of the next throw being heads is 1/2; but the probability that the *next 6* throws are heads is 1/64. (Feel free to try it and keep notes on how often it happens!)
Keith Wild - Probability
Re Roger Fallon's question: It's the very fact that the probability remains 1/2 that reflects independence of each coin toss. Even after 16 throws coming up heads the coin doesn't say to itself "Wow, 17 heads in a row are very unlikely, so I'll make the chance of a head only 1/50 this time." It remains 1/2, i.e. the same as for any other throw and independent of what went on before. The fact that 17 heads in a row is a very unlikely outcome just comes about by multiplying 1/2 by itself 17 times.
C H Collingwood. Probability - The three doors pro
Colva says "You're always better to switch rather than sticking". But suppose a friend had chosen the other door. Would you both have to switch, ie do a swap! Can't both be right! No wonder the others were shaking their heads. The one in three chance became one in two as soon as the third door was opened.
Robert Dobbs - Probability
To answer Roger Fallon's question, firstly a correction: the odds of a head or a tail are the same and they are both 1 to 1 (or 'evens' as they say), not 2 to 1. The use of odds in the context of probability can easily be misunderstood. If someone says the odds are m to n for an event happening then the corresponding probability for that event happening is m/(m+n). But you need to be careful which way round you put the two numbers. A not-for-profit bookie (one not intending to lose or profit himself) might offer you fair odds of 100 to 8 on a nag of a horse. That means he reckons the probability of the horse winning is 8/108.On to independence. Trying to explain this means untangling the words you are using. Your confusion is reflected in the way you express the problem. This is a little tricky for me and may sound rude. However, it is not intended to be. I find it easiest to use an emphatic style. That's all. Please take it in that spirit. You may have to read this more than once as it is a little nit-picky.(i) "[W]e are told that the chances of throwing seventeen heads are very low because each time the odds are 1/2 times 1/2 times 1/2 etc.": that is only 'sort of' correct. If you omit the words "each time" then you would be strictly correct because there is no "each time" here. We'll get to that next. (ii) "This is exactly as we would intuitively consider it with the chances rapidly decreasing": this is where you need to be careful. The chances (i.e. probability) of EACH head are NOT decreasing rapidly — they are always the same, namely 0.5, and they are always INDEPENDENT of each other (I know you know that but it needs saying). But here's the meat: NOTHING is decreasING. However, if we were talking about a SEQUENCE of 17 trials (containing 1 coin, 2 coins, 3 coins, ... 17 coins in each trial respectively) and we worked out the probability for each trial that all the coins in that trial would turn up heads then the resulting SEQUENCE of probabilities (1/2, 1/4, 1/8, 1/16 ...) would certainly be a decreasING sequence. However, we are only talking about ONE trial, not a sequence of trials, and that one trial has 17 coins. There is nothing here that is decreasING. In fact there is NOTHING here CHANGING in any way. The probability is simply 1/2 times itself 17 times.(iii) "So it's not independent?": what is "it" that isn't independent? Each toss is independent of every other toss. There are lots of coin tosses but there is no particular "it" here to be not independent.Incidentally,THHHHHHHHHHHHHHHH is just as likely as HHHHHHHHHHHHHHHHH. And so is HHHHTHHTTHTTTHHHT. There is nothing special about seventeen heads in a row. Any given sequence is as likely as any other.Now, I don't know whether I have helped you but I hope I have.
Heather Gilmour: Probability
Missed the show (looking forward to finding time for podcast tomorrow!) but read the newsletter. Roger Fallon - when you embark on a sequence of 17 throws, 17 heads is a very unlikely outcome - 1/2 times 1/2 times 1/2 etc as you say. Once you have thrown 16 heads in a row (very unlikely - 1/2 x 1/2 x 1/2 etc as before), the probability of throwing another head is.... 1 in 2. Counterintuitive but true. As for the National Lottery, which Melvyn mentions in his newsletter - yes, any six number sequence is as likely to occur as any other - so you might as well choose 1, 2, 3, 4, 5, 6. But I am struggling with the idea that randomness favours clusters, which directly contradicts this, and even more frustrated to realise that this is not going to be in the podcast as it didnt make it past the research notes. Is this really what you meant? I can see that if you choose sequential numbers, and they come up, you are much less likely to have to share the jackpot, because few people would choose a sequence - but dont see how you are more likely to win in the first place?
Paulpic &statistics
The mysterious nature of probability is often used to connect science to the mystries of religion. So then; the old saying, "There are 3 types of falsehoods -lies, dam lies, & statistics.", might make one question orientation of the deity being followed.
Alex Iden. Probability.
Good stuff! Re-sparked my interest in the subject. Better than sudoku for a brain work-out.
Peter Bolt; Proaability
There is a wonderful small hand operated machine in the B`ham Science Museum which links Probability Theory to Clusters in random selection. Based on a design for a bagatell table it shows clusters of ball bearings when "spun".I used it to select my Lottery numbers about 10 years ago. As soon as I win the Jackpot I will let you know.
Ewart Shaw - probability.
This was a fascinating presentation, with a few slips (e.g. "gambler's fallacy" - see Simon Drury's comment below). As a statistician, I also want to point out that I'm just as interested in formal probability as is a pure mathematician, but also need to be aware of the assumptions often unthinkingly made when applying probability to the real world - e.g. for tax & other reasons, babies are not born uniformly throughout the year, so in practice one might need fewer than 23 "random" people to have at least a half chance of matching birthdays. Similarly, if I saw a coin land "tails" 80 times in succession, I would be fairly certain that (a) I was watching a Tom Stoppard play, (b) the coin was double-tailed, or (c) the tosser was cheating. In any case, my probability for "tails" next time would be much greater than 1/2 - see "exchangeability" rather than "independence". It's ignoring things like this that, as was mentioned in the programme, has disastrous practical consequences like Sally Clark's conviction.An interesting follow-up programme would be on "Probability in the Real World", featuring (say) a medical statistician, a quantum theorist, and a philosopher. If they get on to discussing causality as well, expect some fireworks! People may also be interested in the nascent website, understandinguncertainty.org
Percival Andrews - Probability
I believe there was a mistake today in the explanation of conditional probabilities in the example with the game show and three doors. In fact there is no difference between sticking with the original choice or switching to the other door. You can demonstrate this with a small computer program to simulate the situation (I did it with ~1 million trials), and if you think about it, since the game show host will only reveal a goat he is not actually giving you any new information at all!
Graham James - The Birthday Problem
One of the best shows ever - if rather hard to follow in parts. I have always loved the subject and as a kid used to roll a die endlessly and record the results. I taught simple probability for a time and the classic birthday problem was often used to surprise the students. But I wonder if the experts have an answer to this one. On 4th September 1999 I boarded a tram in San Francisco. It was pretty full so there were probaly around 40 persons on it. Right next to where I was standing a lady was sitting down clutching a balloon and birthday cards. Another lady sitting opposite asked her if it was her birthday. The first lady replied that it was indeed her birthday to which the second lady replied that it was her birthday also. Statistically noit a big surprise perhaps but consider this. The 4th September is my birthday too. 3 people out of 40 and all stood or sat adjacent to one another. I am not sure what statistics tells us about that but I shall never forget it. Anyway thanks for a truly great programme. I have downloaded the podcast to listen again.
Max Wallis; Probability and quantum physics
Pure mathematicians stry outside their speciality at their peril. Colver Roney-Dougal said explaining quantum physics (QP) on hidden-variable theories implies "action at a distance" (A-at-D). On the contrary, it's collapse of the wave function instantaneously in QP that implies A-at-D, as Einstein's famous EPR argument. If as in most science since Descartes, A-at-D is ruled out, one has to look for unknown (hidden) structure to explain quantum fluctuations and probabilities. Even a 'simple' electron must have hidden structure, changing in time, for QP gives it up and down spin states. It's the unreal formulation of QP in terms of complex (number) probability, ie. the wave function psi, that should have attracted the criticisms of your experts.
Robert Nield - probability/quantum physics
So, randomness obeys rules - amazing! Yet it isn't. A system is "random" if it obeys the rules of randomness. In maths, "randomness" does not mean absolute anarchy! It is obvious, yet remarkable, that no physical system obeys no rules at all. In other words, existence itself seems to be the manifestation of law. The big questions are: what are the truly fundamental physical laws and what gave them "life"?
Roger Fallon. Probability
I can hazily grasp the idea that after five throws a sixth throw can be said to be completely independent and so, counter-intuitively, the odds are 2 to 1 for heads or tails next throw. However, later we are told that the chances of throwing seventeen heads are very low because each time the odds are 1/2 times 1/2 times 1/2 etc. This is exactly as we would intuitively consider it with the chances rapidly decreasing. So it's not independent? Isn't one coin six times like six coins one time each with the same probabilities?
Les Weston - Trying to cover too much in too short
I found today's programming one of the best I've heard so far. But I do have one major complaint, and that is that many of the topics covered seemed to have been cut short (by Melvyn) before reaching their conclusion.For example, Melvyn interrupted one description (of Jacob Bernoulli's Law of Large Numbers) before we were told how many tosses of a coin would tell us whether it was likely to be biassed, even though it seemed that the interviewee (sorry, I can't remember her name) was just about to move on to that. But the earlier descriptions, of the truncated dice game (Fermat's Last Theorum?) and of Independence of Probability (the multiple coin toss problem), were spot on.
Probability
Dear Mr BraggThe probability of getting an answer to this question is probably minute but I will try all the same.Can one of your experts tell me why it is that an obvious factor is always left out of their assertions? The probability of tossing a coin and getting "heads" is not one in two.There are recorded events all over the world where very accurate throws can result in consistent results of "heads" or "tails and therefore the results are always heavily influenced by another dimension which seems rarely to be voiced. Kind regardsBahman Farzade: bfarzad@proactiveselection.com
Simon Drury - Gambler's fallacy
Your contributor today was wrong - the gamblers fallacy is the belief that if, for example, four reds come up (in a two option system) then black is more likely on the next spin/throw. You had already discussed this before he made his mistake. The 'infinite' doubling of losses that he referred to is actually called the Martingdale betting system, and is not a fallacy but practically not feasible in most situations.
Glenn Williams: 29th March Show
Just wanted to say what a wonderful show. This is really what radio 4 should be all about.
