Beyond Luck & Coincidence: 5 Surprising Probabilities & Odds to Wrap Your Head Around

Beyond Luck & Coincidence: 5 Surprising Probabilities & Odds to Wrap Your Head Around

Some people just seem to be extremely lucky, don’t they? Take Richard Lustig, who has won the lottery seven times, amounting over $1 million in total lotto winnings. Except he claims that his winnings are down to something other than mere coincidence. He’s not suggesting it’s the hand of God or some supernatural force – he says he simply applies a strategy, and it concerns probability.

But if the lottery is truly random, how can there be any truth to his sensational claim?

We’ll take a look at Lustig’s strategy of winning, and dive into four other famous examples of when strategy shouldn’t play a part, yet complex mathematical probabilities mean that they do. Not all methods proposed below can or should be applied in practice, whether on a gameshow, playing the lottery or at an online casino. However they will change your perception of what constitutes true ‘luck’ and true ‘randomness’ – and what is actually the product of a discreet but potentially significant change in odds.

Monty Hall Problem

Marilyn vos Savant, who holds the record of the highest recorded IQ, used to have a column in Parade magazine where she solved mathematical conundrums and puzzles – often ones that had been sent in by readers. In 1990, the so-called ‘Monty Hall Problem’ became famous when a reader posed vos Savant a question (loosely based on the ‘Let’s Make a Deal’ game show premise, whose Canadian host was named Monty Hall).

The reader set up a scenario in which a game show contestant is shown three doors. The contestant knows that behind one door is a prize car. Behind the other two doors, there are prize goats. The contestant gets to pick one door. Let’s say he or she picks ‘Door 1’. They still don’t know what’s behind the door, but now the host will open a different door than the one the contestant picked. It’s important to remember that the host can’t open the same door that the contestant picked, nor can he open the door which reveals the prize car. Monty Hall proceeds to open a door to reveal a goat – let’s say it’s ‘Door 3’.

Now, with Door 3 opened and one goat revealed, the contestant is given the choice of changing their choice of door to Door 2, or sticking with their original choice of Door 1.

Marilyn vos Savant is asked whether changing the initial choice of door to the other unopened door increases one’s chances of winning the car. Intuitively, one would think that it makes no difference – there are two doors left unopened, one of which hides a car and the other of which hides a goat. The chances of winning the car remain 50/50 regardless, right?

Not according to vos Savant, who says that one should switch doors for the best chance of winning the car, and her reasoning has since been proven to be mathematically correct. The puzzle is most easily understood when examining all possible outcomes of switching doors, versus not switching. Take a look at the below table to see how the the contestant has a ⅔ chance of winning the car if they switch from their original choice of door, and only a ⅓ chance if they stick with their original choice.

Behind Door 1 Behind Door 2 Behind Door 3 Result when sticking with Door 1 Result if switching to remaining door
Goat Goat Car Wins goat Wins car
Goat Car Goat Wins goat Wins car
Car Goat Goat Wins car Wins goat

The mind-blowing fact that changing doors does make a difference to your chances was disputed by many prominent mathematicians, some of whom refused to believe the probabilities until shown a computer simulation that revealed it to be true. So, if you ever find yourself faced with this particular situation on a game show – be sure to switch doors!

Monty Hall Problem

Bertrand’s Box Paradox

Here’s another probability paradox where the answer is not what it seems. Imagine you have three boxes, each containing two coins. In one box, the two coins are gold. In a different box, the two coins are silver. The final box has one gold coin and one silver coin. These boxes are shuffled at random, and you are asked to pick one.

You pick one box and, without look, fish a gold coin out of it. Now – what are the chances that the other coin in the same box is also gold?

Many people instinctively answer 50/50 – just like in the Monty Hall Problem. After all, there are two boxes which could have resulted in you picking a gold coin out of them, so the chances must be equal? That is not correct! You have a ⅔ chance of discovering that the second coin is also gold, and only a ⅓ chance of discovering you picked the box with both a gold and silver coin.

This is because the fact that you selected a gold coin plays into the probability. The box with two gold coins was always going to reveal a gold coin first, and there are two ways in which this could have happened (you fish out one gold coin, or you fish out the other gold coin).

The box with only one gold coin, on the other hand, could have equally revealed a silver coin. The fact you picked a gold coin veers toward the likelihood that you in fact picked the box that always reveals a gold coin. Confused?

Box with only gold coins

Coin revealed Gold Gold
Coin left in box Gold Gold

Box with silver and gold coin

Coin revealed Gold Silver
Coin left in box Silver Gold

Box with only silver coins

Coin revealed Silver Silver
Coin left in box Silver Silver

As you can see, one box will ALWAYS show you a gold coin, whereas the other only has a 50/50 chance of showing you gold. So if you pick a gold coin first, it’s more likely that you’ve picked the one that shows gold 100% of the time rather than the one that reveals gold only 50% of the time.

Bertrand’s Box Paradox

Richard Lustig’s Lottery Tips

Richard Lustig is an American man who came to prominence after multiple lottery wins and – in particular – his claims of having a lotto winning strategy. With his book Learn How to Increase Your Chances of Winning the Lottery, he purports that there are ways of increasing your chances of winning the lottery based on how you pick your numbers and how to spend the money you win on the lottery. Lustig’s total lottery winnings amount to just over $1 million with his biggest win coming in at $842,000 in 2002 when he won the Florida Mega Money Jackpot. His smaller wins include $10,000 from Millionaire Magic while in 2010 he won $98,992.92 on Fantasy 5, with five other considerable wins in between.

According to Lustig, there are a few things that you can do to improve your chances of winning lottery jackpots. For example, he claims that players should never go for ‘quick picks’ and always choose their own numbers.

However, Lustig’s methods have been subject to a lot of scrutiny – especially this particular point about choosing your own numbers. Statistics actually show that 70% of all lottery wins come from quick pick lottery tickets (however this could also be because the quick pick option is simply a lot more common than the option of choosing one’s own numbers). There is absolutely nothing to suggest that personally picked numbers are better than quick pick tickets, since lotto results are entirely random. Only if you believe yourself to have an intuition for lucky numbers should you pick your own.

What’s more – Lustig’s advice could even be counter-productive. Quick picks, whilst not increasing your chances of winning the lottery overall, could increase the chance of you getting a bigger prize sum than if you pick your own numbers. Why?

Because people tend to pick their own numbers based on significant dates – birthdays, anniversaries etc. As such, the most commonly picked numbers tend to be lower than 32, meaning you could end up splitting the prize with people who picked the exact same number. Quick picks, which are truly random and more often include higher numbers, reduce the chance of a split lottery jackpot.

Another criticism is pointed toward Lustig’s claim that re-investing all your winnings back into the lottery (i.e, buying more lottery tickets) is a sound strategy. Lustig has never clarified whether he is in fact a ‘net winner’ when it comes to the lottery. From a probability perspective, there’s no denying that the more lottery tickets you buy, the higher your chance of winning, but to re-invest all winnings in an attempt to win again is illogical. Wouldn’t you just be playing the lottery forever rather than spending or investing your money into other things?

Our verdict is that there really isn’t any practical way to increase your odds in the lottery. Richard Lustig has most likely won seven times thanks to a combination of good luck and impractical mass-purchase of lottery tickets. However, by ignoring Lustig’s advice and choosing ‘quick pick’ numbers, you could increase that chance – if you win – you win bigger.

Richard Lustig’s Lottery Tips

Martingale System

The Martingale System is a class of betting strategy that originated from 18th century France and is based on the law of large numbers probability theory. This is a theorem that describes the result of performing the same experiment many times – for example, playing roulette online and betting on red continuously. The strategy involves the player doubling their bet every time they lose – thereby having their first win recover all preceding losses. The theory is that the player will eventually win a profit equal to, or more than, the original stake.

There are many who are convinced by this strategy – but there’s a reason they’re not millionaires. There are two problems to this system. Firstly, if you are playing a casino game like roulette, the house edge means that when playing over longer periods of time, the expected value becomes negative. Secondly, this system will only work if the bettor has infinite wealth, which of course nobody has. Therefore, the Martingale Strategy will lead to the eventually bankruptcy of anyone who takes it seriously. As such, the Martingale system is not the fail-proof, genius strategy some believe it to be. A lot of Live Casino games have a maximum betting range imposed, in any case. When it comes to roulette and many other casino games, strategy is rarely of use. Luck, not strategy, is your friend when it comes to winning big in roulette.

Martingale System

The Probabilities of Slot Machines

Here’s an interesting question; does winning once on a game of ‘chance’, such as slot games, decrease your likelihood of winning again? Surely the chances of winning two, three, four times in a row make the chance of every subsequent win increasingly slim? Yes and no. First of all, due to the Random Number Generator with which all slots must be equipped for fair play, you have exactly the same chance of winning with every spin irrespective of previous wins.

However, even though your odds of winning are the same with each spin, if you consider how likely you are to win again not from the perspective of the ‘specific process’ but of a ‘broader pattern’ – the answer changes.

Statistically, people are less likely to win big again right after winning big once on a slot machine, simply because it is uncommon to observe this. Based on the statistical likelihood in this sense, a winning streak can fairly be described as ‘ good luck’ (unless you believe in Jung’s theory of meaningful coincidences, in which case maybe there’s a great meaning behind your series of wins!). You can also turn this around, and say that the longer you go without winning, the statistically more likely you are to win. Is it useful? Probably not – but slot machines are supposed to be about fun, and not carefully calculated odds. Just rest assured that your chances are exactly the same as everybody else’s with every spin of the reels.

The Probabilities of Slot Machines

A final note on randomness…

Finally, it should be made clear that the above probabilities and odds are based on the underlying assumption that each process and game is entirely fair, and that no small variations or discrete change in condition has an effect upon the outcome. With online slots and indeed most games at an online casino, this is easy to ensure – the RNG is truly random and, since it is digital, not subject to faults or malfunctions that might present themselves in land-based games.

However when it comes to live roulette or baccarat – is it possible that tiny factors could interfere with true randomness? We’re talking about chaos theory, or ‘extreme reliance upon initial conditions.’

What is Chaos Theory?

Chaos theory is the idea that small, unnoticeable (and often impossible to measure) factors can have a dramatic effect on an overall outcome of something – at least over longer periods of time. You may have also heard of this referred to as ‘the butterfly effect’ – the idea that the flap of a butterfly’s wings could cause a tornado elsewhere in the world. Our examples won’t be quite this dramatic – and easier to wrap your head around. Let’s take the roulette wheel.

The player’s odds of winning when betting on a roulette wheel are calculated based on a perfect model where the spin is different each time. But roulette wheels are spun by a human, and humans are imperfect. The haste and angle at which a dealer spins a roulette wheel could influence its likelihood of landing on certain numbers, while the slightest dent in the roulette wheel or tilt in its base cause create a dramatic bias that one might be able to exploit.

When it comes to games like baccarat and blackjack, meanwhile, a close examination of the dealer’s shuffling style would allow for card counting, which naturally increases the player’s odds of winning. Indeed, this is how the MIT Blackjack team were able to make their fortune through a sophisticated and far-spanning card counting scheme. So while these casino games are ‘random’ – they are also not immune to influences which, though minor, could alter the course of the game.

What is Chaos Theory?

How do I use this to my advantage?

Unfortunately, this knowledge – though fascinating – is almost impossible to apply in any practical sense. In live blackjack and baccarat, the deck of cards are switched each time, and dealers change regularly. As for games like roulette, in-depth data would be required to establish whether the wheel has any bias through landing on certain numbers more often than others. By in-depth data, we mean statistics from hundreds of thousands of spins on a specific wheel, as well as stats on what numbers come up more frequently with that specific dealer. It would also necessitate a personal inspection of the wheel, and ideally also overhead footage of its spins. In a land-based casino, such scrutiny is naturally not permitted.

So, whether or not each spin of the roulette wheel or deal of hand in blackjack truly offers the same probabilities each time is impossible to say. For the best chance of winning one should always play under the assumption that odds remain the same for each round. After all, live casinos, bingo halls, and national lottos all go to immense lengths in ensuring that chaos theory doesn’t come into play, and that games remain fair and chance-based. Machine card shufflers, regular examination of game tables, frequent changing of lotto balls are all just a few examples of the precautions they take. As for online casino games like mobile slots; this is where you can sit back, place your bet, and rest assured that you have just as much chance as any other player to win the giant jackpot.