​Every year we have an office sweepstake for the Cheltenham horse racing festival. Like most sweepstakes, this one attempts to remove the skill, allowing everyone to take part without necessarily needing to know much about horse racing.

In case you’re not familiar with a Sweepstake, here’s a simple example of how one based on the World Cup might work:

• Entry costs £5
• Each player randomly selects a team
• The player who picked the team that goes on to win, gets all the money

This allows anyone to enter and have an equal chance of winning (the chance is equal prior to drawing, but not equal once the draw has been made), otherwise everyone would just want Spain or Germany, or whichever team is currently considered the favourite.
 
Note that in order for this to work properly, you need to ensure that the team draw is not carried out until everyone who wants to play has put their money in – otherwise you introduce some optionality and people can then decide whether to enter based on the teams that are still left. i.e. if you know that Germany and Spain have already been picked then there is less value in entering the competition.
 
The rules for our Cheltenham sweepstake were as follows:
 
The Rules
 
The festival comprises 7 races a day for 4 days, for a total of 28 races.
 
The sweepstake costs £20 to enter, and the winnings are calculated as follows:
 
3rd place in competition = 10% of pot
2nd place in competition = 20% of pot
1st place in competition = 70% of pot
 
Each participant picks one horse per race each morning. Points are then calculated using the following scoring system:

• 1st place – 5 x odds
• 2nd place – 3 x odds
• 3rd place – 1 x odds

 
A running total is kept throughout the competition, and the winner is determined after the final race.
 
The odds the scoring are based on are set using the odds printed in the Metro Newspaper on the morning of the races.
 
(as an example, for a horse which has odds of 11/2 in the Metro - if the horse then places 1st, if we selected this horse, we would win (1+11/2)*5 = 32.5 points)
 
Initial thoughts
 
Any set of betting odds can be converted to an implied probability of winning, these would be the odds which over the long run would cause you to breakeven if the race were repeated multiple times with each horse winning a proportion equal to its probability of winning.

Because the scoring in our sweepstake is based on the betting odds, using implied probabilities derived from the odds to help select our horses ends up cancelling itself out (which was the intention when designing the rules). The implied probability can be calculated as one over the odds As an aside, the bookie is indifferent to whether this is the correct probability of winning, they structures the odds purely on the ratio of how their customers are betting. They then structure the odds so that they make money on each race, regardless of which horse wins, for an explanation of this, see my post on creating a Dutchbook:
​
​www.lewiswalsh.net/blog/archives/12-2017
 
Here is an example showing how we would calculate the implied probabilities using some made up odds:
 

​

We can then use the implied probabilities we just calculated to see what would happen if each horse finished in the first three positions. Once we have done this, we can then calculate the Expected Value of betting on each horse:

We see that the payout varies for each horse, but the EV is the same. This is by design, the intention is that it should not matter which horse you bet on, the sweepstake rules equalise everyone.

So what can we - an actuary who knows nothing much about horse racing - do? I don’t have any special knowledge that would allow me to select a horse which would beat the odds listed in the Metro, we appear to be at an impasse.

I could attempt to build a model of which horse will win, and then select my horse based on that, but unless it proves to be more accurate than the Metro odds, I might as well just pick at random. Furthermore, if I could build such a model, then I could just start betting actual money. This probably shows you what a difficult problem this is! There's no such thing as free money. It would be a cool project to try, and it’s something I’ve been meaning to attempt for a while, but that’s best saved for another day.

Attempt 1 - Metro vs pre-race odds

My first thought was that we can exploit the difference in odds between those published in the Metro in the morning, and the latest odds published closer to the start of the race. It seems reasonable to assume that the odds just before the race should be more accurate than the metro odds. There will have been time for additional information to be included in the more up to date odds, e.g. the weather is worse than expected therefore horse x is expected to be slower than usual. Since the payout will be based on the Metro, we will then be able to maximise our EV by exploiting this differential. 

Our table will end up looking something like this:
​

 
We see that we have very small variations in the EVs for some of the horses. It looks like according to this analysis Horse 3 would be the best selection as it has the highest EVs for 1st, 2nd, and 3rd. Based on this strategy, we would then go through each race and select the horse with the highest EV.

Is this what I did? No, for a couple of reasons. The biggest issue was that the Metro did not publish odds in the morning for all races, meaning we couldn’t use the Metro, and therefore the rules of the sweepstake were amended to use the official pre-race odds to calculate the payout instead. This meant there was only one set of odds used, and our edge disappeared!
​
Even if we had used this method, there was a more fundamental issue - the margins we ended up with were tiny anyway. The Metro vs pre-race odds did not swing wildly, meaning that even selecting the horse with the highest EV were only marginally better than picking at random.

So, was there an alternative?

Attempt 2 - 2nd and 3rd place odds
​
My next attempt at an exploitative strategy was based on the insight that the payout multiplier for 2nd and 3rd place was based on the odds of the horse coming 1st, rather than the odds of the horse coming 2nd or 3rd.

The expected value of a horse was not quite as I calculated above, it was actually:
$$EV = P(1)*p_1 + P(2)*p_2 + P(3)*p_3$$
Above, we were using the implied probability of the horse coming first as a proxy for the probability it would come second and third. This is not the same, and some betting websites do allow you to bet on whether a horse will come 2nd or 3rd.

For websites that do not allow you to bet directly on this, then we may still be able to calculate it from the odds for whether a horse finishes in the top 2 or 3 places. We just need to subtract out the implied probability of coming 1st from the probability of coming in the top 2, and then subtracting this out from coming top 3 etc.

​I therefore added some more columns to my table above, corresponding to the probability of the horses coming 2nd and 3rd, and then used this to calculate the EV instead.

We see that the final column, Total EV, now has quite different values for each horse. In this case Horse 15, Seddon has an EV of 11.72. The favourite horse on the other hand - number 7  - only has an EV of 6.2. The intuitive explanation of this is that the probability of Seddon coming first is very low – this is the reflected in the long odds of 67, this then gives us a large multiplier, but the odds of the horse coming second or third are actually relatively less far out – the fact that it is not the favourite actually increases the odds of it coming in a position which is not 1st. But then if does come in 2nd or 3rd, we would still apply the same large multiplier for the odds of it coming 1st. This then gives us our 2nd edge – we can gain a positive EV by focusing.

As a thought experiment, imagine we have a race with three horses – horse A is a clear favourite, horse B is an average horse, and horse C is clearly the weakest. By betting on horse C – the odds of it winning should be very low, so the multiple should be very high, but then this multiple will be applied even if it comes 2nd or 3rd, which is exactly where it is expected to finish.

This therefore suggests our next potential strategy – select horses which maximise our EV using the implied probabilities of the horses coming 2nd, 3rd etc.

So is this what I did? Well kind of....

The issue with this approach is that typically the horses that provide the best EV also have very long odds. In the race analysed above, our horse has an EV of 11.7, but it only has a 7% chance overall of coming in the top 3. In race two for example, the horse with the best EV actually only had a 2.7% chance of coming in the top 3. Since there are only 28 races in total, if each horse we selected only had a 2.7% chance of coming in, then the probability of us getting 0 points overall in the entire competition would then be:
​
$(1-2.7 \%)^{28} = 48 \%$​

So there is roughly a 50% chance we will get 0 points!

Alternatively, if we selected the favourite every time, we could expect it to come top 3 almost every time, and thereby guarantee ourselves points most races, but it also has the lowest EV.

So we have what appears to be a risk vs reward trade off. Pick outsiders and give ourselves the highest EV overall, or pick the favourites thereby reducing our overall EV but also reducing our volatility.

This leads us  neatly to attempt 3 - trying to think about how to maximise our probability of winning the competition rather than simply maximising EV for each race.
​
Attempt 3 - EP curves
​
From the work above, we now have our model of the position each horse will finish in each race – using the 1st, 2nd, and 3rd implied probabilities - and we have the payout for each horse – using the odds of the horse coming 1st. We can then bring our selection of horse and these probabilities together in a summary tab and simulate our daily score stochastically using a Monte Carlo method. To do this we just need to turn the implied probabilities into a CDF, and lookup the value of each position and repeat 10,000 times. The output for this then ends up looking like the following, where the value is the number of points we will for a given race.

So we see that across this sample of 20 simulations, most days we to end up with 0 points overall, but a few days have very high scores. So far so good!

The next step is to set up an EP table of the following, which looks like the following:

The EP table gives us the probability of exceeding various scores in the competition based on our horse selections. In this case, we see that there is a 1 in 20 chance of getting 453 points or greater in the day.

This is useful even on its own – when I was deciding which horses to bet on, I simply played around with the selections until I got an EP table I was comfortable with. My reasoning was quite basic – I decided I wanted to maximise the 1 in 20 value. I wanted to give myself something like a 1/4 chance of winning the whole competition and a 3/4 chance of getting very few points. Since there were four days of races, dividing the 1/4 by another 4 suggests we should be looking at maximising the 1 in 20 level (I admit this reasoning was a bit rough, but it seemed to serve its purpose)

The insight here is that the payout structure of the sweepstake is such that coming in the top 3 is all that matters, and in particular coming 1st is disproportionately rewarded. To see this, we can think of the daily selection of horses as attempting to maximise the EV of the overall prize rather than the EV of our overall score - maximising the EV of each race is only a means to this end. So we are actually interested in maximising the following:

$0.7 * P(1st) + 0.2 + P(2nd) + 0.1 * P(3rd)$

Which will largely be dominated by P(1st), given the $0.7$ factor.
​
This is largely the strategy I went for in the end.

Attempt 4 - Game theory?

I’ve brushed over one difficulty above; in order to maximise our prize EV we need to consider not just which strategy we should take, but how this strategy will fare against the strategies that other people will take. If everyone is maximising their 1 in 20 return period then there’s little point us doing exactly the same. Luckily for me, most people were doing little more than picking horses randomly. We could then formalise this assumption, and come up with a numerical solution to the problem above.

To do this, we would simulate our returns for each day across 10,000 simulations as above, but this time we would compare ourselves against a ‘base strategy’ of random selection of horses and we would simulate this base strategy for the approximately 40 people who entered.

Each simulation would then give us a ranking we would finish in the competition, Here is an example of what that would look like:

​And we could then convert this into an EP table which would look like the following:

So we see that if we select these horses, we end up having somewhere between a 1 in 10 and a 1 in 5 chance of winning the competition.

Now that we have all of this set up, we can then optimise our horse selection to target a particular return period.

I didn’t actually end up setting up the above for the sweepstake, but I suspect it would have been an improvement on my approach

Attempt 5 - Multiple day?

There is a further refinement we can make to the above. We have so far only really been focusing on maximising our chance of winning by using a fixed strategy throughout the competition. But there is no reason we have to do this. After the first day, we should really be including the current scores of each competitor as part of our calculation of our ranking. i.e. if person 1 had a great day and now has 200 points but we had a bad day and still have 0 points, by accounting for this, the model should automatically increase our volatility i.e. start picking horses with longer odds so as to increase the chance of us catching up. If on the other hand, we had a really good first day and are now in the lead, the model should then automatically reduce our volatility and start selecting the favourites more often to help safely maintain our lead.
​
How did it work in practice?

I ended up placing 2nd, and taking 20% of the prize pot which was great! I was behind for most of the competition but then pulled back on the last day when a 66/1 came in 1st place, and I picked up 330 points off a single race. This may look miraculous, but is exactly how the model is supposed to work. 

Does that have any applicability to gambling generally? Unfortunately not, basically all of the work above is based on exploiting the specific scoring system of the sweepstake. There's no real way of apply this to gambling generally.

When you are playing a drinking game, do you ever catch yourself calculating the probabilities of various events happening? If you are like most people the answer to that is probably "..... um..... no....... why would I ever do that...."

Okay, maybe that's not the kind of thing you think about when playing a drinking game. But I bet you think it would be interesting to know what the odds are anyway? No? Really? Still not that interested? .... Okay... well I find it interesting, so I'm going to write about it anyway. 

Eyes up/Eyes down

I've only played this game a couple of times but it's pretty fun, it requires no equipment, and has quite a lot of drinking. All players sit in a circle, one player calls out "Eye's down" at which point everyone looks down, the same player then says "Eye's up" at which point everyone looks up at a random person in the circle. If the person you are looking at is also looking at you, then you both drink. Pretty simple.

What is the probability that you will drink? Let's denote the event that you have matched someone with $M$. Then:

$$P(M) = \frac{1}{n-1} $$
Where $n$ is the number of players.

​Of course people in real life tend not to pick a random person to look at, and even if they attempt to pick a random person, people have been shown to be terrible at picking randomly. But for the purposes of this analysis, unless we make this assumption, the only alternative would be to play the game hundreds of times, record how often matches are made, and then make an empirical claim. As fun as it sounds to play this game hundreds of times in a row, it would be better for your health to just assume a uniform spread of guesses.

The fact that you have a $ \frac{1}{n-1} $ chance of getting a match means that the more people are playing the game, the less likely each person is to drink. Does this apply to the group as a whole though? What is the probability that someone in the group drinks? If we had a hundred people playing, then each individual would hardly ever drink. In fact you would expect them to drink once every $99$ goes. But would you expect it to also be unlikely that anyone at all drinks?  I spent about an hour trying to calculate this and didn't get very far.

Calculating through conditional probabilities doesn't seem to help, and I couldn't come up with a decent approach to count the permutations of all the possible ways of selecting pairings for elements in a set, such that there are no 2-cycles. In the end I gave up and just wrote a program to calculate the answer stochastically. Monte Carlo analysis really is such a versatile tool for problems like these. Anytime you can formulate the problem easily, but struggle to come up with an analytical solution, Monte Carlo analysis will probably work.

Here is the table I generated of the probability of someone in the group drinking for a group of size n:

There is a definite trend here, if we plot the values on the chart it looks pretty clear that out solution converges to $0.6$. No idea why though! I might come back to this problem another time, but if anyone has any thoughts then please let me know.
​

Odds

So this is not technically a drinking game, but is played quite often by people who are out drinking. The rules for this game are pretty simple, anytime you would like to dare someone to do something you say "odds of you doing this". The dare might be anything, from 'eating a spoon full of chilli powder, downing your drink, or even getting a tattoo (that last one is a real one I heard about from a friend who playing the game while travelling in Eastern Europe). The person you have challenged then gives you a number. They might say $20$, then each of you count down from $3$ and say an integer from $1$ to $20$ (or whatever number they selected). If you both said the same number, then they have to do the dare. Therefore the higher number you pick, the less likely it is that you will have to do the dare. 

What are the odds of you having to do whatever it is you are betting on, based on you selecting the number $n$? This is quite a straightforward problem, the answer is just $\frac{1} {n} $. I was playing this game last week with someone who thought the answer might be $\frac{1} {n^2} $. This would give a very different likelihood of having to do the dare. For example if you selected $20$, then you would have a $5 \%$ chance of having to do the dare, but according to my friend's calculation he thought you would have $0.25 \%$ chance of doing the dare.

Here is a table showing the correct probabilities for various values of $n$.

Dirty Pint Flip

I wouldn't recommend playing this game to be honest. I played it in uni once, but it's a bit grim. All the players sit in a circle with a central pint glass in the middle, the players take it in turn to pour any amount of their own drink into the central pint glass as they would like, they then have to flip a coin and guess whether it will be heads or tails. If they get it right, then the game moves on to the person to their left, if they get it wrong, then they have to down the dirty pint. When I played it some people were drinking wine, some people were drinking beer... it was not a good combination.

What is the probability that you will have to drink? This is quite straight forward, it is simply the probability that you will guess the coin flip correctly. Which is just $\frac{1}{2} $.

What about the question of how much you will drink on average? That is, on average if you have to drink, how many people will have poured their drink into the glass before you drink? We can model this as a geometric distribution and calculate the probabilities of each possible number of people. Let's denote the number of people who have poured in before you, given you are about to drink as $N$, then:

The expected value of the number of drinks is then the sumproduct of these two columns . This gives us a value of $2$. Therefore, if you have to drink, the average number of drinks that will have been poured into the pint is $2$.

Two Dice Nominate

This is another game I've only played a couple of times. And I'm not sure if it has a better name than the one I've given it, if you know of one then let me know. In this game, you sit in a circle with a range of drinks on the table, you take it in turns to nominate a person, a drink, and a number between $2$ and $12$. You then roll two dice and add them together. If you have rolled the correct number, the person you nominated drinks the drink you selected, if however you roll a double, then you drink that drink. If it is both a double and you get it correct, then you both have to drink.

The reason that this game does not have much depth to it is that you can always just pick the most likely number to come up when rolling two dice. Here is a table showing the probability of each value coming up:

Therefore you might as well pick $7$ every single time! Is there anything else we can say about this game? We know that if we roll  a double, then we have to drink the drink. What is the probability of you rolling a double in this game? The probability of this is always just $\frac{1}{6} $. Interesting this is the same probability as rolling the dice such that the sum is $7$. Therefore, if you do pick $7$, there is an equal chance of you and the person you are nominating drinking the drink.
​

Recently I've been reading The Mathematics of Poker (2009, Bill Chen and Jerrod Ankenman) and I came across an interesting idea that I thought I'd write about. For me, understanding how to analyse this situation really gets to the heart of how I think about poker. I'd love to spend more time playing and studying poker but it's such a time-sink and I don't really have the oodles of free time it would require, but every now and again I'll still open up a poker book and read about something, this is one of those interesting topics I was reading about hopefully you find it as interesting as I do.

Calling a shove pre-flop in heads up

The scenario being analysed in the book is a relatively common situation, particularly in online poker where people are more inclined to shove than in real life games. The question is:

How should we analyse whether to call when we have a moderately strong hand against an opponent who has gone all-in pre flop.

Let's set up an example so that we have something concrete to talk about. Say we have pocket Kings pre flop, and our opponent goes all-in, how should we decide whether to call? Obviously without knowing our opponent's hand there is no 100% correct answer. There is  however one very useful way of analysing the situation.

Equity against a range

We need to ask ourselves - what cards would are opponent go all-in with, and how does my current hand fare against that range? i.e. we need to calculate our equity against our opponent's range. We are adding another layer of stochastic uncertainty on the event, instead of trying to guess what hand our opponent has (which is almost impossible) we are trying to guess what kind of hands he might go all-in with (which is very much possible). We then take this extra level of uncertainty and calculate the correct mathematical way to proceed.

On the one extreme, let's suppose that based on our read of how our opponent is playing, we might think that they would only go all-in with very strong hands, in this case just pocket Aces. We would then be a 4:1 underdog if we call with Ks, and we should definitely fold. (In order to calculate this we could use any standard poker calculator like the following)

www.cardplayer.com/poker-tools/odds-calculator/texas-holdem

One the other hand, suppose we know for a fact that our opponent has not looked at their cards at all but has still decided to go all in. In this case we should definitely call. The only cards we will be behind are pocket Aces, which make up a small fraction of the possible hands that our opponent could shove with, and we will be ahead or equal against all other possible hands. Therefore we would have a positive EV when calling.

What if our read on our opponent's range is somewhere in between though?

What we need to do is calculate our equity against each individual hand in our opponent's range, and then calculate the probability of our opponent having a given hand from that range. That is to say, in order to calculate the conditional expectation, we need to calculate the unconditional expectations against each hand and then multiply by the conditional probability of our opponent having that hand, given our belief about our opponent's range. 

Numerical Example

Let's go back to our numerical example, and suppose that we have pocket Kings, and we put our opponent on either Pocket Aces, Kings, Queens, or Jacks. All of these hands are equally likely, so there is a 25% chance of our opponent having each hand. We can look up our equity against each hand (after you've been playing for a while, you naturally start to memorise hand equities anyway)

Our probability of winning is then:

$$P(A) * P(beating A) + P(K)*P(beating K) + P(Q)*P(beating Q) + P(J) * P(beating J)$$
Putting in our values:

$$ 0.25*0.2 + 0.25*0.5 + 0.25*0.8 + 0.25*0.8 = 0.575.$$

​We therefore see we have a positive expectation against this range, and should call.

No one actually thinks like this in real games?

It is a general misconception that professional poker is a game where players are trying to guess exactly what hand their opponent has, are constantly trying to bluff each other, or trying to pick up on subtle tells or signs that their opponent is or isn't bluffing.

The more mundane truth is that poker is ultimately a game of imperfect information, where the better player is the one who can correctly interpret the betting information their opponent is giving them, and can then quickly and accurately make the kind of judgements described above during a game.

Obviously poker players are not carrying out these calculations in their head to multiple decimal places in real time, what they will do though is review their hands after a game, calculate exactly what they should have done, and try to build up an intuition as to what the correct answer is, so that in the middle of a game they can quickly make decisions.

Software to Analyse this situation

Is there an easy software based method way of calculating our equity against a range? After I did a quick google there are a few programs that offer this type of analysis. For example:

combonator.com/
www.power-equilab.com/

More interestingly though, I also found the following open source software, that can be adapted to carry out this type of analysis:

github.com/zekyll/OMPEval

At some point, I might try to use this code to set up a page on this website to let people analyse this situation.

Over the Christmas break, I finally got around to playing some Computer Games, which I unfortunately haven't done in ages. One really fun game (if you're like me and love difficult puzzles) is a game called Myst.

It was originally released in 1993 as a point and click puzzle adventure game, but it has subsequently been remade, using the Unity engine, as a full 3d interactive game. I really liked the fact that there is no time limit, no danger of dying, just a cool story, interesting puzzles and relaxing music.

One puzzle that I thought was cool involved aligning gear wheels. If you haven't played the game and don't want spoilers, then stop reading now.

The Gear Puzzle

​We are given the contraption in the image below, and are told that we need to end up with the numbers 2,2,1 (reading from top to bottom). We have three levers we can pull, the left lever moves the top two gears left once, the right lever moves the bottom two gears right once. The lever on the back wall resets the machine. The starting values of the gears are 3,3,3. So how many times do we need to pull each lever in order to end up at 2,2,1?

After playing around with it for a while I began to think that it might not even be possible to get to 2,2,1 using the two levers. Then I thought if that's true, can I prove it as well?

Proof that the puzzle is impossible

Instead of thinking of the value as $3$, let's call it $0$ instead. We can then think of the starting state as a vector:

To see that we are not able to do this with just the basic operations from the starting point we are given, note that the the sum of the first and third elements of our starting vector is equal to the second vector mod $3$, that is, $0 + 0 \equiv 0$ $(mod$ $3)$. For the vector we trying to create, the first and third is not congruent to the second element mod $3$. Instead we have $ 2 + 1 \equiv 0$ $(mod$ $3) $ rather than $2$.

The two operations we are applying to our starting vector preserve this congruence property, as it applies to both the vectors we are adding on as well. Therefore, we can conclude that this property is an invariant of the system, and because our goal does not satisfy this condition, we will never be able to reach:

by just using simple pulls of the lever. We need to realise this in order to force us to look for an alternative way to solve the puzzle. Otherwise we would probably just keep yanking on the levers in the hope that we will eventually get there. Once we realise this and start looking around, we are able to find another way to manipulate the machine, allowing us to make the numbers we need.

I always found it quite interesting that prior to the 19th century, Probability Theory was basically just a footnote to the study of gambling. The first time that Probability Theory was formalised in any systematic way at all was through the correspondence of three 17th century mathematicians, Pierre Fermat (famous for his last theorem), Blaise Pascal (famous for his wager), and Gerolamo Cardano (not actually famous at all) when analysing a problem in gambling called the problem of points. 

The problem of points is the problem of how to come up with a fair way to divide the winnings when betting on a game of chance which has interrupted before it can be finished. For example, let's say we are playing a game where we take it in turns to roll a dice and we record how many 6s we get, the first person who rolls a total of 10 6s wins. What happens if we are unable to finish the game, but one player has already rolled 8 6s, whereas their opponent has only rolled 2 6s. How should we divide the money in a fair way? Obviously it's unfair to just split the money 50-50 as the player with 8 6s has a much higher chance of winning, but at the same time, there is a chance that the player with only 2 6s might get lucky and still win, so we can't just give all the money to the player who is currently winning. The solution to the problem involves calculating the probability of each player winning given their current state, and then dividing the money proportionally. In order to answer this question in a systematic way, Fermat, Pascal, and Cardano formalised many of the basic principles of Probability Theory which we still use today.

Newton - Pepys Problem

The Newton - Pepys problem is another famous problem related to gambling and Probability Theory. It is named after a series of correspondence between Isaac Newton and Samuel Pepys, the famous diarist, in 1693. Pepys wrote to Newton asking for his opinion on a wager that he wanted to make.

Which of the following three propositions has the greatest chance of success?
A. Six fair dice are tossed independently and at least one “6” appears.
B. Twelve fair dice are tossed independently and at least two “6”s appear.
C. Eighteen fair dice are tossed independently and at least three “6”s appear.

Pepys initially believed that Option C had the highest chance of success, followed by Option B, then Option A. Newton correctly answered that it was in fact the opposite order and that Option A was the most likely, Option C was the least likely.

Wikipedia has the analytical solution to the problem. Which comes out as:

We can clearly see from this graph that Option A is the most likely Option of the three, with Option C being the least likely. We can tell all of this by just setting up a model that takes 5 minutes to build and give an answer in seconds. It makes you wonder what Newton would have been able to manage if he had access to the computing power that we take for granted now.

Sources:
Wikipedia: en.wikipedia.org/wiki/Newton%E2%80%93Pepys_problem
An Article by Stephen Stigler: arxiv.org/pdf/math/0701089.pdf

I've been playing quite a lot of Poker recently, and the more Poker I've played, the more I've come to realise that it's actually a very deep game. One of the interesting features of Poker that makes it so complex is the inability to see each other's cards (this may sound like I'm stating the obvious but bear with me). Let's consider a thought experiment, what if we played Poker against each other without hiding our cards? I show below why it would still be a fairly difficult game to play well. The fact that we also can't see each other's cards  just adds another level of uncertainty to the underlying complexity. 

Incomplete Information
​
In technical terms Poker is a game of incomplete information. An example of a deep game which has complete information is Chess. In Chess we can see all our opponents pieces, and they can see all our pieces. The fact that we have perfect information when playing Chess in no way makes the game trivial.

Heads Up No Limit Texas Holdem

Let's suppose we are playing heads up No limit Holdem (No-limit Texas Holdem is the proper name for the game most people think of when they think of Poker, heads up just means that there are only two of playing). Let's first consider the case where we can't see each other's cards.

Suppose we have J♣6♣, and the flop is 10♣ 9♣ 8♠ there is $£100$ in the pot, we have a stack of $£200$, and our opponent also has a stack of $£200$. Our opponent goes all-in, what should we do?

Obviously there is no 100% correct answer to this question, and if we were playing in real life, we'd have to consider a large number of other variables. How likely do we think our opponent is to bluff? What was the betting pre flop? Do we think we are a better player than our opponent overall, and therefore want to reduce the variance of the game? Are we the two short stacked players in a tournament?

All of the above considerations are what make poker such an interesting game.

Playing with Complete Information

Now let's suppose that we can see our opponent's cards. For example, imagine we are in the same situation as above, but now our opponent shows us his cards, they have 5♥ 5♣. What should we do now?

It turns out in this case, there is actually a mathematically correct way of playing, but it is by no means obvious what that is.

At the moment our opponent has a better hand - they have a pair of 5s, and we do not have anything. What we do have though is the possibility of making a better hand.

We can reason as follows. We will win if one of the following happens:

• A Queen or 7 of any suit come on the turn or the river.
• A Club of any value comes on the turn or the river. 
• Two consecutive Jacks of any suit come on the turn and then the river.
• Two consecutive 6s of any suit come on the turn and then the river.

We can calculate the probability of one of these happening, and we also know how much we will win if one of them occurs, we can therefore calculate the expected value of calling our opponents all-in.

It turns out that, the probability of us winning this hand if we call is $68.38$%, the probability of there being a split pot if we call is $1.82$%, and the probability of us losing the pot if we call is $29.08$%. In order to see how to calculate this, we note that there are $46$ cards remaining in the deck, and that we can calculate the probability of getting the cards we need for the $4$ way of winning.

Now that we have the probability of winning, we can compare this to the pay off should we win.

Expected Value of calling

The Expected Value is the average amount we would expect to win over a large sample of hands if we make a given decision.

In this case we calculate the expected value as:

$EV = 0.6838 * 300 - 0.2908 *200 = 146.98$

Therefore, over a large enough number of hands, if we are presented with this exact situation repeatedly and we take the average of how much we win or lose, we would expect the amount we win to converge to $£146.98$

The thing about this I find interesting is that when you are playing in real life, even when you can see your opponent's cards, unless you have access to an EV calculator it is still very difficult to determine the correct way to play, and this difficulty is only compounded by the fact that we also can't see our opponent's card.

In order to play Poker well when playing with Incomplete Information, we have to attempt to reason about who has the better cards, whilst bearing in mind this underlying uncertainty which is present even when playing with Complete Information.