What's the edge in Satoshi Mines 25 Squares 3 Bombs

In my last post, i showed you how to calculate the true odds for the bitcoin game satoshi mines

If you want to know how the game works, see this post.  Or if you want to play the game go here

In that post, i showed you the true odds of the game (25 squares and 3 bombs) and in my previous post, i showed you the payout odds (for 25 squares and 3 bombs)

I thought it might be worth comparing them....   The table below shows true odds vs actual odds (for 25 squares 3 bombs)

I think the interesting thing here, is the cumulative edge isn't a constant.   They offer a very low edge for your first bet and then the edge gets progressively worse until your 8th guess when the edge starts reducing.   I don't have their payouts beyond guess 16.   I might create a robot to get those numbers at some point.   Let's look at the edge changes as a graph.

So what does this tell me.......

It says to me you really shouldn't play beyond your first guess... for 25 squares 3 bombs
As the cumulative edge (unto guess 17 doesn't get better than that first edge on that first guess..

You'd be better to cash out your money, and then just play another round.   You'll have lower odds sure (as you'll always been playing with 22/25 probability) but you'll be playing with a lower edge, so you'll lose money over the long run.

It also tells me there maybe more value in the other variants of the game and we'll explore that in another post.

In true bitcoin tradition, if you found this article useful and want to donate me some spare bitcoins my wallet address is:  17QCjNaaXUWyG5WWaV3KVDQG7HdVkSDLs1

In the next article we'll look at some of the grids and options you have

Calculating the true probability of Satoshi Mines (25 squares, 3 bombs)

In my last post, i explained how to play a bitcoin game called satoshi mines

In this post, i'm gonna explain how you can calculate the probabilities of the game.   I'll start with 25 square, 3 bomb grid.

If you need a refresher on calculating probabilities, i'll do one here but you can always refer to this post.

If you want to play the game, you can play satoshi mines here

Calculating the probability

It's pretty simple,.   There are 25 squares, 3 of which have bombs.   

First guess

That means the probability of me NOT HITTING a square on my first guess is....

22/25 or 88.50% (or decimal odds of (1.136)

Yes there are 22 squares out of 25 squares that are empty.   So I have an 88.5% (22/25) chance of not hitting a bomb in my first attempt.

If I had an original stake of 100 bits, i'd need to win over 13.6 bits for me to be able to beat the game,  as you could see from my previous post.   The actual payout is 13 bits.  So the house has an edge of about 0.5% (which is actually pretty low for a game)

Second guess

So it's a little harder for me now.  Now there are 21 squares (i clicked a square remember) that are empty and 3 squares which have bombs.  So I a smaller chance of hitting an empty square.  So the probability of me not hitting a square in my second attempt is.....

21/24 or 87.5% (or decimal odds of 1.143 roughly)

So now that I am staking 113 bits (initial stake of 100 plus my 13 bits from last win), i really need to win around 16 bits for this to be fair to bring my cumulative wins to 129 bits.   Well you've probably figured that isn't gonna happen.   The win is 15 bits bringing my stake / win unto 128 bits

So how is cumulative probability calculated.

Basically for each guess, it is

probability of current guess * probability of previous guesses

so for 2 guesses, it'd be

(22/25) * (21/24)

for 3 guesses, it's be

(22/25) * (21/24) * (20/23)

etc etc

Anyways, hopefully now when you're playing Satoshi mines, you'll be able to calculate the true probability.   i've put a handy chart below for you to see the true probabilities of the 

25 square 3 bombs game true probabilities 

In the next posts, i'll look at the probabilities of the various combinations   And look at giving you, your best strategy.

In true bitcoin tradition, if you found this article useful and want to donate me some spare bitcoins my wallet address is:  17QCjNaaXUWyG5WWaV3KVDQG7HdVkSDLs1

Hope this helps

How does Satoshi Mines work?

So I've spent some of time over the last few days looking at some of the bitcoin gambling sites.   Wow there is a lot to talk about in this subject.   Anyways I found myself playing a fun little game called Satoshi Mines, it's kinda like Minesweeper.

It's Bitcoin only so if you want to gamble, you will need a bitcoin wallet (a subject for another day) or    you can play in a practise mode.   I was playing in practise mode and recommend you do the same because I'll prove in my next post, that you can't really win.

How does the game work?

You start with a square (default is 25 squares) with a hidden number of bombs (default is 3).  You can change the number of bombs if you want (and i recommend you do).   Here is a screenshot.

You bet an initial stake (let's say 100 bits) and everytime you click on a square and miss a bomb, you win some bits.   You can either cash out your bits or risk those bits you just won against the chance of wining some more bits.  Obviously if you hit a bomb, you lose all the bits won (and your original stake).

What's the payouts?

I played the game for a little bit and captured the following payouts for a 25 square game with 3 bombs

As you can see the more squares you uncover as empty, your cumulative odds get better and you can win a good multiplier of your original stake.  As you can see above, the best I did was to manage 16 squares where i could win 23 times my stakes (if I won and cashed out).

So in the above example.

• My initial stake was 100 bits
• If i hit a mine, i lose all 100 bits.
• If I've guessed 10 times (i.e. picked 10 squares that don't have bombs in them), i'll have 466 bits (366 bits won plus my original 100 bits)
• If i want to guess another square (11th guess), i'll either win another 111 bits, making a total of 577 bits or i'll lose everything including my initial 100 bits stake.   Or I can cash out and take my 466 bits as winnings.

Okay, looks fun, can I win?   In the short term, sure.  In the long term, no.  Why?   Probability!   And that's what i'll explain in my next post

However the game does have some pretty low house edges (as i'll explain in my next post)

Warning:  This is a not a casino, not licensed and you're playing with Bitcoins.  I really do recommend you only play in practise mode.

Hope this helps

High-Odd-Black vs High-Odd-Red in Roulette

So I read this interesting article at

http://www.roulette30.com/2013/08/an-observation-on-even-chances.html

Which basically says you have a higher probability of losing if you bet High-Odd-Black than you do if you bet High-Odd-Red.

So the idea is that if I place a bet on High Numbers, Odd Numbers and Red Colors then there is only 5 numbers that can make me lose:

2, 4, 8, 6 and 10

But if I bet on High, Odd and Black then there is theoretically only 4 numbers that can make me lose

12, 14, 16 and 18

And then the article did some math wizardry showed that with each bet being evens then the "probability" of losing all your bets would be 1/8 (1/2 * 1/2 * 1/2).   BTW, the article ignores zeroes.  So i will too.

However, he points out if we were to bet on High-Odd-Black the probability of losing all your bets is
4/36 (ignoring zeroes) which simplifies to 1/9  but..... if we bet on High-Odd-Red the probability of losing all your bets is... 5/36 or 1/7.2

So should I bet on High-Odd-Black instead of High-Odd-Red?

Errrr, no......   Because you don't win bets on how many unique numbers lose.   These are not combinator bets.    If I was offered odds of 1/8 for all my bets to lose then sure this is a winning strategy and I'd say take the house to the cleaners.   As 1/9 will yield profit on 1/8 you'd have your own edge.

That's not the bet.   The bet is 3 separate bets whose results are paid out in isolation   This is not a trixie bet, this is 3 individual bets.   And sometimes you will win more than once for a certain number.   What we really should be looking is how many times can I win across these bets.

Well, here we go....

High-Odd-Black

Oh, look, we have 54 possible wins.    Shall we see how that breaks down

• 14 numbers offer as a single win (only one bet wins (High, Odd or Black)
• 14 numbers offer as a win on two bets (High and Odd, High and Black or Odd and Black)
• 4 numbers off a win on all threes bets (High, Odd and Black)

High-Odd-Red

And we still have 54 possible wins.  And shall we see how that breaks down also

• 13 numbers offer as a single win (only one bet wins (High, Odd or Red)
• 13 numbers offer as a win on two bets (High and Odd, High and Black or Odd and Red)
• 5 numbers off a win on all threes bets (High, Odd and Red)

What does this mean?

Don't get me wrong, it's an interesting observation but it's not gonna help you win more.   the odds of winning with High-Odd-Black are still the same as High-Odd-Red (But i applaud the observation skills)

It just means, that if we bet High-Odd-Black then sure we have a greater coverage of winning numbers than High-Odd-Red but we'll win less.

It all evens out.   We're betting individual bets.  And as shown they have a different distribution.   But at the end of the day.   There are only 54 possible winning bets.....  regardless of which way you bet

And just in-case you wanna check out the math.  here is a spreadsheet with my breakdown.

https://www.dropbox.com/s/fwcgk8eb0171lus/Kavouras%20HighOddBlack.xlsx?dl=0

Hope this helps

Hot numbers in roulette

I'm not gonna name the site but I was looking at yet another person peddling their roulette strategy based on tracking numbers (no genuine advantage play).

Basically, you stick in the last 6 numbers and voila, it will tell you where to bet.   If the wheel spins 3, 4, 5, 14, 15 and 17.   You should probably bet on 19-36, cause those numbers are due....

Just for fun, let's see how that looks in the Excel Simulator

The Random Numbers

This is what my Excel Random Number generator produced (feel free to replicate this experiment yourself)

3, 14, 9, 9, 3, 21, 35, 3, 29, 0, 19, 36, 4, 9, 5

The distribution

Oh look the number 3 came up 3 times (surely that's a hot number)????  BTW, that was sarcasm.   We know enough about random numbers now to know that that can easily happen.  However this could be an interesting blog post in the future.   Hot numbers.

What might be more interesting is the ranges, which is what they were betting on in this system.

Their system was either avoid betting on 1 to 12 because it's hot.  but it might have been bet on 1 to 12 because it's hot.

There is no hot....  It's just some random numbers.   nothing is due or not due....

The kicker

And here was the kicker.  Don't worry, if you lose, just martingale and it'll recover your losses.

And we know how that ends...

Anyways, you know my thought on this sort of thing already.  I could type up their system into Excel and formally disprove it.   But you can always do that yourself if you want.

We know enough about random numbers, streaky numbers and martingale to know this is a silly strategy.

For more info on Martingale, read my martingale post (and feel free to download the excel tool)

For more info on streaks, read my post on streaky coin tosses

For generating roulette numbers in excel, read my post (or look at the martingale one)

Why playing Red or Black in Roulette using Martingale is a bad idea

I've been leading unto this post for a bit now.

One of the most popular techniques in roulette is to use a progressive betting pattern such as

• Martingale
• Reverse Martingale
• d'Alembert
• etc

In all honesty, these are all a pile of rubbish.  If the math was corrupt, do you really think the casino's would let you play the game?   So the idea of this post is to let you see how much of a pile of rubbish they are.

So what is Martingale

The basic idea is that everytime you lose a bet, you double up your previous bet until you recoup your losses. Here is an example with a £10 stake and an initial bank of £10,000

Oh, wow, how amazing.  this must be free money.   Except, if you hit a long streak of losses the next stake is gonna exponentially grow.

And this is how much you'll have to bet to recover your losses during a losing streak

Oh wow after a losing streak of 11 losses, I'm betting £10,000 to recover my money.   Or 15 losses, I'm betting £163,000.   Get's pretty uncomfortable pretty quickly.

What else, can go wrong?  Well, even if you are stupid enough to try and bet £20,000 to recover your losses, the casino probably won't let you.  They have table limits (i.e. a maximum bet size).   And the chances are it's somewhere between £500 and £10,000.   

Yeah but i'll never get a streak that long

What's the chances of hitting a streak of 10 losses in a row and busting the table limits.   Errrr, pretty high.   See my post on streaky coin tosses (more favourable than red or black on roulette)

As you can see from the above graph (which is 20,000 simulated coin tosses).  You're gonna hit a streak of 10 or more, 18 times in 20,000 coin tosses.   In roulette this will work out basically around 1 in every 1,000 games.   That's pretty frequent.

Still don't believe me.

Try it yourself.   I've created a martingale simulator in excel.  Which uses my excel roulette simulator and an enhancement of my level staking plan simulators

The simulator allows you to simulate a large number of random roulette runs.   You can put in whatever table limits you want (it handles it) and you can generate new runs.

And as you can see in the above run, i bust out after about 1,000 games.   Sure i had a maximum bank of 3350 but i eventually bust out.

In-fact, you always bust out. 

Some people will say, ahhh but you play it until you win and then go home.   Sure but what happens the next night.....?    It's a small amount of games until you bust out.   You can't win.

Try it, play it in my simulator.   You can't win

Download my martingale simulator

You can download it from my dropbox for free.

https://www.dropbox.com/s/sa3nyie9wadxvkf/Roulette%20Martingale%20Simulator.xlsx?dl=0

Play with it until your heart is content.  (please note the simulator doesn't handle half-bet returns and doesn't need to the point is to show why martingale doesn't work)

You can even modify if you wan to try out your own wacky staking plans

So, next time you're tempted by playing martingale, just stop and think about it before you throw away your money

Hope this helps

Simulating Roulette in Excel

In one of my earlier posts, i showed you how to simulate a coin toss in excel (or google docs)

In this article, we will look at extending that technique to allow us to simulate roulette.   We're obviously just gonna randomly throw a roulette number and not simulate physics or mechanics (for now ;).   The idea is that you will be able to use this simulator to test out any crazy betting systems you may come up with.

This technique will work for both European and American wheels.   So let's get to it.

Setting up our wheel

First we want to do is setup our roulette wheel in Excel, and it should look something like this...

• First column is a list of numbers from 0 to 36 (37 if you want to do an American Wheel)
• Second column is the corresponding number on the roulette wheel (following actual layout)
• Third column is the color of the number

So to set it up, do the following

1) Open Excel
2) Rename the tab to "European Wheel"
3) In the first row enter the headers (as shown in the screenshot): Number, Actual Number Color
4) In column A enter in each row the numbers 0 to 36
5) Enter the corresponding number in the roulette wheel in order in Column B
6) Enter the color of the actual number in Column C

European Wheel

The following table gives the layout of a European Wheel, you can just copy and past it into excel

Number	Actual Number	Color
0	0	Green
1	32	Red
2	15	Black
3	19	Red
4	4	Black
5	21	Red
6	2	Black
7	25	Red
8	17	Black
9	34	Red
10	6	Black
11	27	Red
12	13	Black
13	36	Red
14	11	Black
15	30	Red
16	8	Black
17	23	Red
18	10	Black
19	5	Red
20	24	Black
21	16	Red
22	33	Black
23	1	Red
24	20	Black
25	14	Red
26	31	Black
27	9	Red
28	22	Black
29	18	Red
30	29	Black
31	7	Red
32	28	Black
33	12	Red
34	35	Black
35	3	Red
36	26	Black

American Wheel

Setting up an American wheel is also pretty easy.  Just create a new table and name it "American Wheel" and set it up the same as a European wheel (0..37 though) and paste in the following layout

Number	Actual Number	Color
0	0	Green
1	28	Black
2	9	Red
3	26	Black
4	30	Red
5	11	Black
6	7	Red
7	20	Black
8	32	Red
9	17	Black
10	5	Red
11	22	Black
12	34	Red
13	15	Black
14	3	Red
15	24	Black
16	36	Red
17	13	Black
18	1	Red
19	00	Green
20	27	Red
21	10	Black
22	25	Red
23	29	Black
24	12	Red
25	8	Black
26	19	Red
27	31	Black
28	18	Red
29	6	Black
30	21	Red
31	33	Black
32	16	Red
33	4	Black
34	23	Red
35	35	Black
36	14	Red
37	2	Black

Setting up the random number generator

So we have our wheels setup, so we just need to setup our random number generator.   Which we want to look something like this

In this diagram:

• Column A shows our randomly roulette number
• Column B shows the color of the number corresponding to it's color on the European Wheel

This is pretty easy to setup

• Manually enter the headers (Winning Number, Winning Color)
• In row 2, enter the following formula in Column A

=VLOOKUP(RANDBETWEEN(0,36),'European Wheel'!$A$2:$A$38,1)

This basically generates a random number between 0 and 36 (RANDBETWEEN(0,36).   And then looks up the corresponding number on our European Wheel tab.   If we want to do this for an American Wheel, we would change the tab to American Wheel and set RANDBETWEEN to generate numbers between 0 and 37,

• Enter the following formula in Row 2 of Column B

=VLOOKUP(A3,'European Wheel'!$A$2:$C$38,3)

This will look up the color of the generated number in our European Wheel (again if you want to simulate American Wheels then switch the tab to American Wheel.

• Do the draggy extends paste thing to copy it to multiple rows

Voila, one very simple roulette simulator in Excel.

This is very simple and easily allows you test betting plans etc.   I use it for disproving systems such as martingale.   It usually proves it to fail pretty quickly.   I obviously have more advanced versions which can do billions of rows of testing but Excel isn't so suitable for that.    However this should be pretty suitable for your needs.

In a future post, i'll show you how to test staking plans such as Martingale and Reverse Martingale with the simulator.   So you can see for yourself how ridiculous these staking plans are.

Hope this helps

UPDATE

You can download a copy of my simulator from my dropbox

https://www.dropbox.com/s/3btgpurbtsuof50/Roulette%20Simulator.xlsx?dl=0

True odds vs Real Odds for Roulette

In my last post, we discussed how you calculate the probability of a single event.

In this post we'll show you how to convert that to odds and then compare the real odds to the probable odds.  In this example we'll look at Roulette.

Red or Black Probability

For now we'll focus on European Roulette (although the theory works just as well for American Roulette).    In the last post we calculated the probability for Red or Black to be 18/37.  In this example we'll bet Red.

There are 18 outcomes that can win for us (i.e. 18 reds), specifically the numbers

1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 35

And 19 outcomes that will lose for us

0, 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35

How to convert probability to Decimal Odds

The true probability for Red or Black is 18/37.  

To convert this to odds we need to do the following conversion

1 / probability

so for Red or Black this would be

1 / (18 / 37)

which would make the real odds

2.0556

The casino is offering odds of 2/1 which is equivalent in decimal odds of

2.0  (it's 2/1 ;)

What does this mean?

This means the casino is offering odds lower than the true probability of winning.   And the casino will profit on the game on average by 2.7% (I'm ignoring "en prison" for now).   Remember over a long period of time the actual results will be close enough to the real probability (see other posts).   So the casino can expect to make 2.7% on average for every game.

So if that roulette table stakes $100,000,000 per year.  Then that roulette table will be profiting by over $2,700,000

And this is how casinos make money.

The bad news is.......

If you play $100,000 worth of games on red or black.   you'll probably lose just over $2,700

We play 10,000 games with $10 stakes.

We win 4864 games (10,000 * (18/37)
We lose 5136 games (10,000 - 4864)

We win 4864 * 10 = $48,640
We lose 5136 * 10 = $51,360

In total we lose (£2,720), which the casinos bank as profit

The odds are in the casinos favor...

In future posts, we'll show:

• Your bankroll degrading using an excel roulette simulator and how the casino's always win
• A full post on odd conversions
• How to manage books / be a bookie / be a casino
• How fancy betting systems don't help when there is no advantage (Martingale, Reverse Martingale, Fibonacci, James Bond) - they'll all lose you money and we'll show it.

Hope this helps

Calculating Probability for single events

In the last post we figured out how to calculate the number of possible outcomes for coin tosses (or any game).  Now we'll look at the probability.

We can calculate the number of possible outcomes using the following formula

number of combinations ^ (number of tosses)

otherwise known as

number of combinations to the power of number of tosses.

So to calculate the probability of an event, we are basically saying the following

the number of times the desired outcome can occur / the number of possible outcomes

Coin Toss (2 sided coin)

So for a 2 sided coin (heads or tails).  there are 2 possible outcomes.

H, T

So there is 1 in 2 chance of flipping a head (as shown in blue below)

H, T

1 instance of the desired outcome within a set of 2 possible outcomes.  Giving a 1/2 chance (50% chance),

Coin Toss (3 sided coin)

So for a 2 sided coin (heads, tails or body).  there are 3 possible outcomes.

H, T, B

So there is 1 in 3 chance of flipping a head (as shown in blue below)

H, T, B

1 instance of the desired outcome within a set of 3 possible outcomes.  Giving a 1/3 chance (33.3333333% chance),

Dice Throw (6 sided dice)

So for a 6 sided dice throw there are 6 possible outcomes.

1,2,3,4,5,6

So there is 1 in 6 chance of rolling a 3 (as shown in blue below)

1,2,3,4,5,6

There is also a 1 in 6 (16.6666666%) chance of hitting any specific number

And there is a 3 in 6 (otherwise known as 1 in 2 (or 1/2 or 50%)) chance of hitting an even number

1,2,3,4,5,6

And the same for hitting an odd number

1,2,3,4,5,6

Specific Number in European and American Roulette

Now that we know how to calculate probability.  Let's look at roulette (we will have bigger discussions on this later).

However in European Roulette, we have the following numbers (note the number zero)

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36

So in European Roulette, the probability of predicting a specific number is 1 in 37, i.e. 1 desired outcome in 37 possible outcomes (2.7% chance)

Specific Number in American Roulette

And in American Roulette, we have the following numbers (not the extra number, the double zero)

0,00,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36

So in American Roulette, the probability of predicting a specific number is 1 in 38, i.e. 1 desired outcome 38 possible outcomes (2.63% chance)

Red or Black in European Roulette

In European Roulette, the zero number is green.   And every other number (1 to 36) is evenly split between red and black (will explain the exact wheel in another post).   That means there are:

18 reds in 37 possible outcomes

18 blacks in 37 possible outcomes

1 zero (green) in 37 possible outcomes

Which means if you were to bet red or black, you would have an 18 in 37 (18/37 or 48.6% chance)

Red or Black in American Roulette

In American Roulette, the zero and double zero numbers are green.   And every other number (1 to 36) is evenly split between red and black (will explain the exact wheel in another post).   That means there are:

18 reds in 38 possible outcomes

18 blacks in 38 possible outcomes

2 zero numbers (green) in 38 possible outcomes

Which means if you were to bet red or black, you would have an 18 in 38 (18/38 or 47.36% chance)

Okay, so now you know how to calculate probabilities in single events.

In future posts, I'll discuss the following

• Calculating Odds from Probabilities
• How to make a book and calculating edge and over round
• Calculating probabilities for sequenced events and multiple outcomes (double dice throws, lottery numbers etc).

Calculating the number of possible outcomes of a Coin Toss

In the previous posts, we've been concentrating a lot on coin tosses.

I thought it might be useful for us to look at how we can calculate the possible number of outcomes.  

In order to calculate the probability of an event to occur mathematically (or to be able to effectively analyze what happened, we need to be able to calculate all possible outcomes).

So in the case of a coin toss.   There are always two possible outcomes in a coin toss.   You will either flip heads or tails.   So let's look at how this breaks down for multiple coin tosses.

1 coin = 2 outcomes =  H,T
2 coins = 4 outcomes = HH,HT,
                       TH,TT
3 coins = 8 outcomes = HHH,HHT,HTH,HTT,
                       THH,THT,TTH,TTT
4 coins = 16 outcomes = HHHH,HHHT,HHTH,HHTT,HTHT,HTTH,HTTT,
                        THHH,THHT,THTH,THTT,TTHT,TTTH,TTTT

And so on and so forth.   And it grows quite quickly.   Here's how the growth looks charted in Excel.

Hey doesn't the graph look like the streaky graph we did the other day.  Sure it's a typical binomial distribution (we'll discuss that another day).  For now we'll concentrate on how to calculate on the number of outcomes for higher number ranges.

In our run of 39 coin tosses, it would probably take us a long time to figure out how many outcomes there were if we were to calculate each and every outcome.   So we wanna take a shortcut.

Inferring the number of outcomes for a coin toss

Just by looking at the coin toss data, we can probably infer that the number of outcomes doubles every time we add a coin.   2 outcomes, 4 outcomes, 8 outcomes, 16 outcomes...

And you'll probably notice the pattern as well.   If you look at 3 coin tosses,  you can see that the first line is the exact same sequence as 2 coin tosses except the first coin toss is a head.   And the second line is again the same sequence as 2 coin tosses but with a tail as the first coin toss.    So sure, it makes sense that it doubles up all the time.

So we are really number of combinations (heads or tails) * number of combinations for previous number of coins

1 coin = 2 outcomes

2 coins = 4 outcomes = 2 outcomes * 2 combinations

3 coins = 8 outcomes = 4 outcomes * 2 combinations

4 coins = 16 outcomes = 8 outcomes * 2 combinations

What would happen if we had a 3 sided coin

Let's say for a second our coin has 3 sides, (Heads, Tails and errrrr Body).  What would happen?

1 coin = 3 outcomes =    H, T, B

2 coins = 9 outcomes =   HH, HT, HB, 

                         TH, TT, TB, 

                         BH, BT, BB

3 coins = 27 outcomes =  HHH,HHT,HHB,HTH,HTT,HTB,HBH,HBT,HBB,

                         THH,THT,THB,TTH,TTT,TTB,TBH,TBT,TBB,

                         BHH,BHT,BHB,BTH,BTT,BTB,BBH,BBT,BBB

Okay, so now we seem to be tripling up from the previous set of outcomes.

And we're following the same pattern as before.

First line is the exact same sequence as 2 coin tosses except the first coin toss is a head.   And the second line is again the same sequence as 2 coin tosses but with a tail as the first coin toss.    And the third line is again the same sequence as 2 coin tosses but with a body as the first coin toss.  So sure, it makes sense that it triples up all the time.

So we are still number of combinations (heads, tails, body) * number of combinations for previous number of coins

1 coin = 3 outcomes

2 coins = 9 outcomes = 3 outcomes * 3 combinations

3 coins = 27 outcomes = 9 outcomes * 3 combinations

4 coins = 81 outcomes = 27 outcomes * 3 combinations

This is cool but would still be a pain to calculate 39 combinations.   What's the formula for doing this?

Quick way for calculating number of outcomes

The quick way of doing this is calculating this is 

number of combinations ^ (number of tosses)

otherwise known as 

number of combinations to the power of number of tosses.

So for a 3 sided coin with 2 tosses, we would calculate it as 

1 coin = 3^1 = 3 = 3 outcomes

2 coins = 3^2 = 3 * 3 = 9 outcomes

3 coins = 3^3 = 3 * 3 * 3 = 27 outcomes

4 coins = 3^4 = 3 * 3 * 3 * 3 = 81 outcomes

And for our regular 2 sided coin

1 coin = 2^1 = 2 = 2 outcomes

2 coins = 2^2 = 2 * 2 = 4 outcomes

3 coins = 2^3 = 2 * 2 * 2 = 8 outcomes

4 coins = 3^4 = 2 * 2 * 2 * 2 = 16 outcomes

How would we do this in Excel (or google docs)?

Pretty easy really.  Lets generate a table of coin tosses (2 sided coin) with their number of outcomes.  Something like this

To generate this in excel, we just do the following

1) In column A, just put the number of coin tosses to generate.  

2) In column B, just put in the following formula

=POWER(2,A2)

3) Do the cell draggy thing to generate the table.

And if you want, you can generate a chart like I did.

BTW, the answer is there is 549,755,813,888 (around 550 billion) possible outcomes for 39 coin tosses.

And if you don't believe me, try it yourself, you know how to check it now