Roulette Mathematics | The Math of Winning in Roulette (2024)

In this article I'm going to give you a simplified introduction to the mathematics of roulette.

I'm mostly going to be working out the probability of winning when betting on red (an even-money bet) on a European roulette table (the one with a single green 0), but you can use the same mathematics for working out the probabilities of winning with any bet on any table.

Probability

1. What's the probability of winning on one spin?

If you want to work out the probability of winning when betting on the red numbers, you can use a simple fraction like this:

18/37 = 0.48648

In other words, there are a total of 37 numbers on the roulette wheel, and 18 of them are red, so we will win 18/37 times. If we put this fraction in to a calculator we get 0.48648, which is roughly 48.7%.

To work out the probability of other bets you do the same thing: put the total number of winning results over the total number of possible results:

Single Number: 1/37Two Numbers: 2/37Three Numbers: 3/37Six Numbers: 6/37Dozen: 12/37Column: 12/37Red: 18/37Black: 18/37Even: 18/37Odd: 18/37

And that's all there is to working out basic probabilities. Just create a fraction that includes the number of times you can win over the total number of possibilities. This gives you the probability of winning, or "p" for short.

2. What's the probability of winning on multiple spins?

We can take this further by calculating the probability of winning multiple times in a row. We do this by multiplying individual probabilities together:

18/37 * 18/37 = 0.2366

In other words, this tells us the probability of winning two times in a row when betting on red. The probability of winning on each individual spin is 18/37, and by multiplying them together we combine them to find the probability of winning over multiple spins.

Whenever you use the word "and" when working with probabilities, you're looking to multiply those things together.

We can work out the probability across further spins by multiplying even more probabilities together:

18/37 * 18/37 * 18/37 * 18/37 * 18/37 = 0.027249

As you can see, the more times we try and win in a row, the lower the probability gets. That's what happens when you multiply fractions together.

Anyway, we can make the above easier to write down with the following mathematical notation:

(18/37)^5 = 0.027249

This is the same thing as saying "18/37 multiplied by itself 5 times".

It's just easier to write it this way, rather than writing out the same fraction over and over again. So the next time you see it, you'll know it's shorthand for "the probability p of the same event happening over multiple spins x".

3. What's the probability of winning on any spin?

Instead of working out the probability of winning on one spin and another, we can also work out the probability of winning on one spin or another (notice the difference).

We do this by adding individual probabilities together:

18/37 + 18/37 = 0.97297

So in other words, we have a 97.3% chance of winning at least once when betting on red over the course of two spins. We don't care to win both, we just want to know the probability of winning on one spin or the other.

Whenever you use the word "or" when working with probabilities, you're looking to add those things together.

This is interesting, but I'm only going to be multiplying probabilities throughout the rest of this article.

4. What's the probability of winning and losing in a specific order?

(I'm using green and red to represent winning and losing.)

We can use the same mathematics as above to help us to work out the probability of an exact series of results.

For example, we can work out the probability of winning the first two spins and then losing the next three spins:

18/37 * 18/37 * 19/37 * 19/37 * 19/37 = 0.032047

If 18/37 numbers will win, then 19/37 (the rest of the numbers) will lose.

So there's a 3.2% chance of this exact sequence of results taking place over the course of the next five spins of the wheel. This was easy to work out, because we're just multiplying individual probabilities like we did before.

To simplify it we could write it like this:

(18/37)^2 * (19/37)^3 = 0.032047

Again, this just shows the probability of winning two times and the probability of losing three times (in a specific order). Don't be put off by the mathematical notation, we're just using it to give us a quick way of writing down a specific scenario.

So over n spins we can work out the probability of winning x times and losing the rest of the time. For convenience we call the probability of losing "q".

5. What's the probability of winning and losing in any order?

But what if we don't care about the order? What if we want to know the probability of winning any two out of the next five spins? That's much more interesting.

To work this out, we first of all need to figure out all the possible combinations of ways we could win the next two out of five spins:

W W L L LW L W L LW L L W LW L L L WL W W L LL W L W LL W L L WL L W W LL L W L WL L L W WCombinations = 10

So we can see there are 10 combinations of ways we could win two out of five spins.

Therefore, to work out the probability of winning two out of five spins in any order, all we have to do now is add up all the individual probabilities of each of these sequences:

(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047(18/37)^2 * (19/37)^3 = 0.032047Total = 0.32047

So now we know there is a 32.1% chance of winning two out of any of the next five spins. All we needed to do was to work out the probability of winning two out of five spins in an exact order, then find out all the possible combinations of doing this, and add that many of those probabilities together.

To simplify the above mathematics, we could have just written something like this instead:

combinations * [(18/37)^2 * (19/37)^3] = 0.32047

And like I say, for working out the probability of results in any order, you just need to be able to work out all the possible combinations to help you find the answer.

n = number of spinsx = number of winscombinations = n!/[x!(n-x)!]

5 = number of spins2 = number of winscombinations = 5! / [2! * 3!] = 5*4*3*2*1 / [ (2*1) * (3*2*1) ] = 5*4 / 2*1 = 20 / 2 = 10

100 = number of spins2 = number of winscombinations = 100! / [2! * 98!] = 100*99 / 2*1 = 9900 / 2 = 4950

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Binomial Distribution

Binomial: "Bi" means two and "nomial" means terms. The two "terms" we are looking at are the probability of winning and the probability of losing.

Now we can work out all the probabilities for winning 1, 2, 3, 4, 5, etc. out of 100 spins, and see how they all compare.

For example, we can work out the probability of winning 1 out of 100 spins like so:

(18/37)^1 * (19/37)^99

But that's for winning once at a specific point over 100 spins. We want to work all the combinations of ways this could happen, because we we're happy for the win to come at any time:

combinations = 100! / [1! * 99!] = 100

So there are 100 ways we could win 1 out of 100 spins (which makes sense when you think about it, because there are 100 different positions where the single win could take place).

But anyway, multiplying the probability of a specific sequence by all the possible combinations gives us:

combinations * probability of specific sequence = 100 * (18/37)^1 * (19/37)^99 = 0.0000000000000000000000000010757

So pretty slim odds.

Anyway, we can repeat this process to find the probabilities for winning x out of 100 spins. Here are all the results:

0/100: 1 * (18/37)^0 * (19/37)^100 = 0.00000000000000000000000000001135501438101/100: 100 * (18/37)^1 * (19/37)^99 = 0.00000000000000000000000000107573820451122/100: 4,950 * (18/37)^2 * (19/37)^98 = 0.00000000000000000000000005044646001155063/100: 161,700 * (18/37)^3 * (19/37)^97 = 0.00000000000000000000000156118518351535474/100: 3,921,225 * (18/37)^4 * (19/37)^96 = 0.00000000000000000000003586617540023436755/100: 75,287,520 * (18/37)^5 * (19/37)^95 = 0.00000000000000000000065238685359584160606/100: 1,192,052,400 * (18/37)^6 * (19/37)^94 = 0.00000000000000000000978580280393763001517/100: 16,007,560,800 * (18/37)^7 * (19/37)^93 = 0.00000000000000000012449307025761255433718/100: 186,087,894,300 * (18/37)^8 * (19/37)^92 = 0.00000000000000000137106183954765366148229/100: 1,902,231,808,400 * (18/37)^9 * (19/37)^91 = 0.000000000000000013277651498777277563827310/100: 17,310,309,456,440 * (18/37)^10 * (19/37)^90 = 0.000000000000000114467332394722100858571311/100: 141,629,804,643,600 * (18/37)^11 * (19/37)^89 = 0.000000000000000887258748705500724576811412/100: 1,050,421,051,106,700 * (18/37)^12 * (19/37)^88 = 0.000000000000006234160155378128136336192013/100: 7,110,542,499,799,204 * (18/37)^13 * (19/37)^87 = 0.000000000000039979391441777158011080325314/100: 44,186,942,677,323,584 * (18/37)^14 * (19/37)^86 = 0.000000000000235367394728657755038525713315/100: 253,338,471,349,988,800 * (18/37)^15 * (19/37)^85 = 0.000000000001278416586105131924908777861716/100: 1,345,860,629,046,814,720 * (18/37)^16 * (19/37)^84 = 0.000000000006434136107700167430198244903817/100: 6,650,134,872,937,201,664 * (18/37)^17 * (19/37)^83 = 0.000000000030118928157407594765397233887318/100: 30,664,510,802,988,204,032 * (18/37)^18 * (19/37)^82 = 0.000000000131572159845517400584063934964519/100: 132,341,572,939,212,275,712 * (18/37)^19 * (19/37)^81 = 0.000000000537951545517960486661079469683520/100: 535,983,370,403,809,853,440 * (18/37)^20 * (19/37)^80 = 0.000000002064035140434702127136944662549421/100: 2,041,841,411,062,131,589,120 * (18/37)^21 * (19/37)^79 = 0.000000007449149379012454084177808347937522/100: 7,332,066,885,177,660,407,808 * (18/37)^22 * (19/37)^78 = 0.000000025341364633865358755448218787062523/100: 24,865,270,306,254,660,567,040 * (18/37)^23 * (19/37)^77 = 0.000000081417107427796224819224854725441324/100: 79,776,075,565,900,366,872,576 * (18/37)^24 * (19/37)^76 = 0.000000247465155471327968820659422288788025/100: 242,519,269,720,337,158,242,304 * (18/37)^25 * (19/37)^75 = 0.000000712699647757424867840854356554336526/100: 699,574,816,500,972,471,189,504 * (18/37)^26 * (19/37)^74 = 0.000001947660980713609924825811467452574327/100: 1,917,353,200,780,442,800,750,592 * (18/37)^27 * (19/37)^73 = 0.000005057084651677443982293030239061337728/100: 4,998,813,702,034,726,544,474,112 * (18/37)^28 * (19/37)^72 = 0.000012490618857714592997631627979693291829/100: 12,410,847,811,948,290,443,837,440 * (18/37)^29 * (19/37)^71 = 0.000029379023665332339339262279764852792130/100: 29,372,339,821,610,946,892,136,448 * (18/37)^30 * (19/37)^70 = 0.000065870863586481979176472101311645701631/100: 66,324,638,306,863,454,235,918,336 * (18/37)^31 * (19/37)^69 = 0.000140912203937126174167759895539120407232/100: 143,012,501,349,174,321,585,586,176 * (18/37)^32 * (19/37)^68 = 0.000287850258700511076569161783567096790633/100: 294,692,427,022,540,688,947,412,992 * (18/37)^33 * (19/37)^67 = 0.000561927777750279478126416154992739393534/100: 580,717,429,720,889,503,917,277,184 * (18/37)^34 * (19/37)^66 = 0.001049047832766002641066704903494155587435/100: 1,095,067,153,187,963,076,310,204,416 * (18/37)^35 * (19/37)^65 = 0.001874088459136858840145545279654015757836/100: 1,977,204,582,144,933,240,600,788,992 * (18/37)^36 * (19/37)^64 = 0.003205677627470942918364027462985177408037/100: 3,420,029,547,493,939,811,572,514,816 * (18/37)^37 * (19/37)^63 = 0.005253116112157223994982846448920099646838/100: 5,670,048,986,634,688,515,759,144,960 * (18/37)^38 * (19/37)^62 = 0.008250739156767716853124738918268121779039/100: 9,013,924,030,034,634,124,564,627,456 * (18/37)^39 * (19/37)^61 = 0.012426214438532758380095977202017820673140/100: 13,746,234,145,802,810,415,403,499,520 * (18/37)^40 * (19/37)^60 = 0.017952609807248633838616669322618690785041/100: 20,116,440,213,369,981,264,747,036,672 * (18/37)^41 * (19/37)^59 = 0.024889369180781186624562550946393457707042/100: 28,258,808,871,162,573,880,278,646,784 * (18/37)^42 * (19/37)^58 = 0.033123446202994483167358907849120441824243/100: 38,116,532,895,986,721,524,916,682,752 * (18/37)^43 * (19/37)^57 = 0.042326655857926859249928952522168401628744/100: 49,378,235,797,073,732,718,773,141,504 * (18/37)^44 * (19/37)^56 = 0.051946350371092089859281770714005688205445/100: 61,448,471,214,136,243,133,410,705,408 * (18/37)^45 * (19/37)^55 = 0.061242013069077054499000922760387766175046/100: 73,470,998,190,815,045,584,871,751,680 * (18/37)^46 * (19/37)^54 = 0.069370243636597536673882302693527890369347/100: 84,413,487,283,064,058,245,720,047,616 * (18/37)^47 * (19/37)^53 = 0.075507140890003113242734400500921765342448/100: 93,206,558,875,049,914,871,998,054,400 * (18/37)^48 * (19/37)^52 = 0.078984443430990119994206111186940688639949/100: 98,913,082,887,808,056,769,296,465,920 * (18/37)^49 * (19/37)^51 = 0.079408634856505663779380199684965191409050/100: 100,891,344,545,564,237,256,087,044,096 * (18/37)^50 * (19/37)^50 = 0.076733817682391805559127817559783579781751/100: 98,913,082,887,808,056,769,296,465,920 * (18/37)^51 * (19/37)^49 = 0.071269799705007852441696059031528420746352/100: 93,206,558,875,049,914,871,998,054,400 * (18/37)^52 * (19/37)^48 = 0.063623444675928078617666017180454218760153/100: 84,413,487,283,064,058,245,720,047,616 * (18/37)^53 * (19/37)^47 = 0.054588536444887640619594293411864782683654/100: 73,470,998,190,815,045,584,871,751,680 * (18/37)^54 * (19/37)^46 = 0.045011600226486328524266866679681697860455/100: 61,448,471,214,136,243,133,410,705,408 * (18/37)^55 * (19/37)^45 = 0.035664693767971961113616430338879581540856/100: 49,378,235,797,073,732,718,773,141,504 * (18/37)^56 * (19/37)^44 = 0.027150753714339545541456288901827065274157/100: 38,116,532,895,986,721,524,916,682,752 * (18/37)^57 * (19/37)^43 = 0.019855398838187356125306592957713291980358/100: 28,258,808,871,162,573,880,278,646,784 * (18/37)^58 * (19/37)^42 = 0.013945624955314898582581228936305706156459/100: 20,116,440,213,369,981,264,747,036,672 * (18/37)^59 * (19/37)^41 = 0.009404899613040206690639521980301651638060/100: 13,746,234,145,802,810,415,403,499,520 * (18/37)^60 * (19/37)^40 = 0.006088435012652340660999072952108690515261/100: 9,013,924,030,034,634,124,564,627,456 * (18/37)^61 * (19/37)^39 = 0.003782289222700334612586869909023334912562/100: 5,670,048,986,634,688,515,759,144,960 * (18/37)^62 * (19/37)^38 = 0.002253961828807838838062416186858172295663/100: 3,420,029,547,493,939,811,572,514,816 * (18/37)^63 * (19/37)^37 = 0.001287978187890194241294050669921489316064/100: 1,977,204,582,144,933,240,600,788,992 * (18/37)^64 * (19/37)^36 = 0.000705422264091175288923396546891808611765/100: 1,095,067,153,187,963,076,310,204,416 * (18/37)^65 * (19/37)^35 = 0.000370132491604114742707321017434196619466/100: 580,717,429,720,889,503,917,277,184 * (18/37)^66 * (19/37)^34 = 0.000185951730231732268979263911212740367867/100: 294,692,427,022,540,688,947,412,992 * (18/37)^67 * (19/37)^33 = 0.000089397061195459599997406585636383624768/100: 143,012,501,349,174,321,585,586,176 * (18/37)^68 * (19/37)^32 = 0.000041100506462928069984982815121554722369/100: 66,324,638,306,863,454,235,918,336 * (18/37)^69 * (19/37)^31 = 0.000018057888423071373934806690964016695570/100: 29,372,339,821,610,946,892,136,448 * (18/37)^70 * (19/37)^30 = 0.000007576166721860019942469920140037942271/100: 12,410,847,811,948,290,443,837,440 * (18/37)^71 * (19/37)^29 = 0.000003032713142923951815974970350198169972/100: 4,998,813,702,034,726,544,474,112 * (18/37)^72 * (19/37)^28 = 0.000001157219488747297230713992864659100973/100: 1,917,353,200,780,442,800,750,592 * (18/37)^73 * (19/37)^27 = 0.000000420503693099234240235478454045248174/100: 699,574,816,500,972,471,189,504 * (18/37)^74 * (19/37)^26 = 0.000000145351916675837745021780352851215075/100: 242,519,269,720,337,158,242,304 * (18/37)^75 * (19/37)^25 = 0.000000047736629476696203161729064158927476/100: 79,776,075,565,900,366,872,576 * (18/37)^76 * (19/37)^24 = 0.000000014876373451879008022617753895754177/100: 24,865,270,306,254,660,567,040 * (18/37)^77 * (19/37)^23 = 0.000000004392750055510412304868746023083378/100: 7,332,066,885,177,660,407,808 * (18/37)^78 * (19/37)^22 = 0.000000001227124509434083379027252165690879/100: 2,041,841,411,062,131,589,120 * (18/37)^79 * (19/37)^21 = 0.000000000323745040463622040869512135283280/100: 535,983,370,403,809,853,440 * (18/37)^80 * (19/37)^20 = 0.000000000080510279799506071407236832597881/100: 132,341,572,939,212,275,712 * (18/37)^81 * (19/37)^19 = 0.000000000018832813988188549386539940283482/100: 30,664,510,802,988,204,032 * (18/37)^82 * (19/37)^18 = 0.000000000004134032338870657300624899127583/100: 6,650,134,872,937,201,664 * (18/37)^83 * (19/37)^17 = 0.000000000000849350968797776207995040981584/100: 1,345,860,629,046,814,720 * (18/37)^84 * (19/37)^16 = 0.000000000000162845486498821792124686755285/100: 253,338,471,349,988,800 * (18/37)^85 * (19/37)^15 = 0.000000000000029039938149635118933093628586/100: 44,186,942,677,323,584 * (18/37)^86 * (19/37)^14 = 0.000000000000004798520991677768642242599187/100: 7,110,542,499,799,204 * (18/37)^87 * (19/37)^13 = 0.000000000000000731534960618753179120546088/100: 1,050,421,051,106,700 * (18/37)^88 * (19/37)^12 = 0.000000000000000102379892813868568821561889/100: 141,629,804,643,600 * (18/37)^89 * (19/37)^11 = 0.000000000000000013077502571138736365519390/100: 17,310,309,456,440 * (18/37)^90 * (19/37)^10 = 0.000000000000000001514237139816065605523191/100: 1,902,231,808,400 * (18/37)^91 * (19/37)^9 = 0.000000000000000000157641807499648127001992/100: 186,087,894,300 * (18/37)^92 * (19/37)^8 = 0.000000000000000000014609824264841536860893/100: 16,007,560,800 * (18/37)^93 * (19/37)^7 = 0.000000000000000000001190613861990482137994/100: 1,192,052,400 * (18/37)^94 * (19/37)^6 = 0.000000000000000000000083996274698096744395/100: 75,287,520 * (18/37)^95 * (19/37)^5 = 0.000000000000000000000005025815882212989496/100: 3,921,225 * (18/37)^96 * (19/37)^4 = 0.000000000000000000000000247984336293404297/100: 161,700 * (18/37)^97 * (19/37)^3 = 0.000000000000000000000000009687939345157498/100: 4,950 * (18/37)^98 * (19/37)^2 = 0.000000000000000000000000000280960646959599/100: 100 * (18/37)^99 * (19/37)^1 = 0.0000000000000000000000000000053772372624100/100: 1 * (18/37)^100 * (19/37)^0 = 0.0000000000000000000000000000000509422477

Note: All of these individual values add up to 1.

If we put all these results in a chart, we get this:

Note: Some of the numbers are too small to show up on the chart, but they are there.

And there we have it. Now we can see all the probabilities of winning x out of the next 100 spins of the roulette wheel when betting on red (where each of those x wins could come in any particular order). It took a bit of mathematics, but we got there in the end.

For what it's worth, this is called a binomial distribution. The formula for this is:

Which is just a mathematical formula for working out what we have just done.

Code

# -----------------------------# Binomial Distribution Formula# -----------------------------# Allows you to work out the probability of x successes over n trials.## P(x) = n!/[x! * (n-x)!] * p^x * q^(n-x)## n = number of trials# x = number of successful trials# p = probability of success# q = probability of failure## Criteria:# * Each trial is independent.# * There are only two possible outcomes from each trial.# * The probability of success and failure stays the same.# Factorial functiondef factorial(n) # create a range from 1 to n and multiply together (return 1 if n is zero) (1..n).inject(:*) || 1end# Variablesn = 100 # number of spinsp = 18/37r # probability of winning (18 red numbers out of 37 on the wheel)q = 1 - p # probability of losing (all the remaining numbers)# Work out binomial probability for each x number of wins out of 100 (e.g. 0/100, 1/100, 2/100, etc.)0.upto(n) do |x| # Combinations of ways we can win that number of times combinations = factorial(n) / (factorial(x) * factorial(n - x)) # Probability of winning x out of n times (in one specific order) probability = p ** x * q ** (n - x) # Multiply together result = combinations * probability # Print results (converting from rational number and showing 40 decimal places) puts "#{x} #{"%.40f" % result.to_f}"end

Statistics

Now that we've got our probability distribution for winning at roulette, we can use a handful of handy formulas to find out more interesting statistics about it.

Before we get started I just want to make a note of these three variables:

n = 100 <- number of spinsp = 18/37 <- probability of winningq = 19/37 <- probability of losing (same as 1-p)

1. Mean

mean = n*p = 100*(18/37) = 48.648

The mean tells us the average number of times we can expect to win over a number of spins.

So in our example over 100 spins with a probability of winning 18/37 on each spin, we can expect to win an average of 48.648 times.

2. Variance

variance = n*p*q = 100*(18/37)*(19/37) = 24.981

The variance indicates how spread out each of the probabilities are from the average.

This number doesn't tell us much on it's own, because it's not in the same units as "number of wins". So it's not like we're saying "ah yes, 24.981, of course!" Instead, this number is only really useful for comparing the shapes of different probability distributions:

So we can see that the probabilities are most spread out when betting on red/black, and least spread out when betting on single number. In other words, you're more likely to see a range of different results when betting on red/black (i.e. in the range of 40-60 wins out of 100), whereas your results are more concentrated around a fixed point when betting on a single number (i.e. in the range of 1-5 wins out of 100).

The peak of each distribution also helps you to identify the most common number of times you expect to win over 100 spins for each bet.

However, whilst the variance is only useful for describing the spread of results and the shape of the distribution, it's actually a stepping stone toward getting a more useful number in the form of the standard deviation...

3. Standard Deviation

standard deviation = sqrt(n*p*q) = sqrt(100*(18/37)*(19/37)) = sqrt(24.981) = 4.998

The standard deviation (SD) tells us what the "typical" deviation is from the mean. It's similar to variance, but this time the actual number is relevant (because the units are back in number of wins).

For example, when betting on red, the mean tells us that can we expect to win an average of 48.648 times out of a 100, and the standard deviation tells us that we should expect to be within 4.998 wins either side of this on a "typical" run.

Furthermore, there are some useful facts about this standard deviation:

• 68% of the time we expect to be within 1 SD (±4.998)
• 95% of the time we expect to be within 2 SD (±9.996)
• 99.7% of the time we expect to be within 3 SD (±14.994)

So basically, the standard deviation helps us to determine if our results over a certain number of spins is statistically "normal" or not.

Code

# --------------------------------# Binomial Distribution Statistics# --------------------------------n = 100 # number of trialsp = 18/37r # probability of success (a bet on the red numbers)q = 1 - p # probability of failure# 1. Mean (the average expected number of wins)mean = n * p# 2. Variance (the spread of the results from the average)variance = mean * q# 3. Standard Deviation (the typical expected deviation from the average)standard_deviation = Math.sqrt(variance)# Show resultsputs "n = #{n}"puts "p = #{p}"puts "q = #{q}"putsputs "Mean = #{mean.to_f}" # convert from fraction to decimal (from rational to float)puts "Variance = #{variance.to_f}"puts "Standard Deviation = #{standard_deviation}"

Conclusion

The mathematics of roulette is just fractions and multiplication.

The mathematics of roulette isn't too difficult. It's essentially working out the individual probabilities of winning, and multiplying them together to work out overall probability of winning over multiple spins.

I'm hoping that the mathematics in this article was easy to follow. Mathematics can appear daunting at first, especially when you run in to terms like "binomial distribution" alongside complex-looking formulas, but when you peel back the terminology the logic is rarely as difficult as it seems.

Lastly, if you're going to take anything from this article, it should be the fact that you're going to win an average of 48.648 out of 100 spins when betting on red.

What does that mean? It means that you're mathematically expected to lose more times than you win. You can try all sorts of systems and place your bets in all kinds of ways, but this number never changes. It's what gives the casino the edge, and it's why you can never beat the game no matter how hard you try.

But on the other hand, at least working out the math was fun.

Links

• Binomial Distribution:
• The Binomial Distribution (mathsisfun.com)
• Binomial Formula Explained (mathnstuff.com)
• Combinations:
• Combinations and Permutations (mathsisfun.com)
• Statistics:
• Mean, Variance, and Standard Deviation of Binomial Distribution (David Arnold)
• Variance and Standard Deviation (mathbitsnotebook.com)