The law of small numbers is a type of cognitive bias which causes people to believe that relatively few observations have the potential of reflecting the general populace. Please read on and use the hospital quiz to check your logical powers, and to figure out how graphs may possibly mislead people. You’ll also learn how you can avoid losses by effectively using statistics during your bet placements at portals like Bet365 etc.
Quite interestingly, when it comes to casino games like slots machines, it’s not the law of small numbers, but the law of large numbers that does the damage! As per this law, the larger the sample size of a certain event, the more likely it reflects the probability of its happening. So, in various online casino games, specifically in online slots games, the machines are programmed in a way that they return a specific percentage to players over a period of time. This may make a deluded gambler bet huge sums of money on slots, even if he’s well aware of this law, or for that matter its inverse. This law impacts these games regardless of whether you’re playing the best of the new-age slots machines, or the conventional ones.
About the hospital quiz
In the year 1974 two popular psychologists named Amos Tversky and Daniel Kahneman presented the following scenario to their experimental subjects, along with an important question. There are 2 major hospitals that serve a specific town. While 45 babies are born each day in the bigger hospital, the smaller hospital delivers 15 babies on a daily basis.
As we may be aware that around 50% of all newborns are boys, their exact percentage may vary from one day to the other. While some days the percentage of boys may be higher than 50%, it may be lower on the other. Both these hospitals recorded the days when the percentage of newborn boys was higher than 60%, for a time period of one year.
Which hospital will record more number of such days?
– The smaller hospital
– The bigger hospital
– It’ll be almost the same (with a possible variance of 5%)
As per the binomial theory, the total number of days when boys will outnumber the girls by minimum 6:4 are more likely to be 3 times higher in the smaller hospital when compared to the bigger one, simply based on the fact that the former is more likely to witness higher volatility in the birth ratios. On the other hand, when it comes to a bigger sample, it has comparatively lesser chances of straying very far away from 50%.
Nevertheless, no more than 22% of all the respondents were able to give the correct answer.
What is meant by heuristics?
Tversky and Kahneman give a name to this error, and called it an inherent belief in the law of small numbers. Generally speaking, the judgements formed from smaller samples are often viewed as being a representation of the bigger population. To give you an example, if we talk about a small sample that appears to be randomly distributed, it would very easily confirm the belief that the bigger population from which that sample has been derived, would also be randomly distributed.
So, it’s proven by the hospital quiz that the bigger sample has comparatively lesser chances of straying very far from the 50% mark. Still, a large majority of people don’t think that way.
On the other hand, a smaller sample that demonstrates a visibly meaningful pattern, for instance, 18 tails from 20 coin tosses, may make the observers believe that the same visible pattern may be displayed by the entire population. In such a scenario, the person may assume that the coin is actually biased. Apophenia is the name given to the experience of viewing patterns in meaningless or random data.
The gambler’s fallacy’s expression is another important example of representativeness heuristic. Quite obviously, this type of bias originates from a belief in the law of small numbers. As it was explained by Tversky and Kahneman, the gambler’s fallacy’s origin lies in the misconception of the fairness of laws of chance. A gambler may think that since a coin toss is always fair, he can expect any deviation in a particular direction to be soon cancelled by deviations in the other direction. The subjects act in a manner as if all the segments of a random sequence should reflect the actual proportion; you can expect a corrective bias (in the opposite direction) to come into play in the event that a sequence strays away from the popular proportion.
Going through some unequal sample sizes
It is highly likely that sports bettors betting on portals like Bet365 or offline may recognise patterns in a faulty way, owing to their misplaced belief in law of small numbers. (On a side note, if you’re keen on placing bets at Bet365 and wish to learn about the various bonuses offered by it currently, you must pay a visit to). The act of misinterpreting profitability from smaller betting samples as being a representation of departure from the randomness and a sound evidence of predictive skills, can result in negative financial consequences in the long run.
Let’s take the example of a hypothetical profitability chart wherein around 100 bets are placed on the NFL point spreads (at an online casino or at some offline bookmaker), with every bet placed at the odds of 1.95. What if we were to tell you that this record was obtained from a famous sports handicapper? Considering the decent growth trend and the 15% yield, you may be forgiven for believing us!
You’ll get the hang of the bigger picture if you look at a chart of 1000 bets. If you look closely you’ll observe that there was actually no scope of long-term profitability at all in this. Why? It was merely produced with the help of a random number generator which assumed that there was a 50% individual win chance, and a -2.5% profit expectation. The first chart that we talked about above is actually a representation of the first 100 bets (of the second chart).
Still, a healthy profitability percentage was maintained for many hundreds of bets placed in the second example of longer time series. In addition, regardless of depicting an overall loss, the time series pattern seems anything but being random in nature, exhibiting a pretty consistent wavelike pattern.
But, Tversky and Kahneman had recognised that we are more likely to perceive sequences of the same outcomes as being non-random in nature, even if there wasn’t any underlying mechanism in them.
Which of the 2 binary sequences shown below, seems random to you, and which one does not?
1, 1, 1, 1 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0
1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0
Majority of people would go for the second sequence as being more random in nature. To tell you the truth, it was the first one actually which was generated randomly in MS Excel! The second one was simply made up purposely, to show shorter sequences of 0s and 1s to you! We’re prone to switching from 0 to 1 and vice versa whenever we need to create random sequences such as this one, in case we feel that either 0 or 1 is occurring a little too often.
Now if we go over some 1000-bet charts which were all randomly generated, we'll know that how easy it is for us to get fooled by seemingly meaningful patterns whenever we are faced with a large range of possible outcomes.
Please keep in mind, we’re talking about 1000 bets and not a series of 100 bets here. We can make use of binomial distribution for working out the chances of being profitable after a certain betting period, regardless of having a -2.5% expectation.
Number of bets (50% winning chances & 1.95 odds) / Minimum required wins / Profit probability
10,000 / 5129 / 0.51%
5000 / 2565 / 3.4%
2500 / 1283 / 9.68%
1000 / 513 / 21.46%
500 / 257 / 28.05%
250 / 129 / 32.9%
100 / 52 / 38.22%
We enjoy more than 1-in-5 probability of being profitable even after having placed 1000 bets, regardless of the fact that our bets are nothing more than being random in nature. It’d have taken us almost 4 seasons if we were to place one handicap bet on each and every NFL game (at a portal like Bet365 etc.). That’s a fairly long time to come to terms with the fact that there’s something else other than luck that may be helping us.
So, how small is too small?
This type of cognitive bias (the law of small numbers) is one which causes people to exhibit a tendency of believing that your observations are better reflections of the general populace. Furthermore, as is revealed in the above exercise, small can often be quite large! Why so because people have a tendency of favouring certainty over doubt, causation over association, skill (specifically self-serving skill) over chance, patterns over randomness and explanation over ignorance. It can be very expensive for sports bettors betting at some offline or online casino, if they don’t truly appreciate the significance of this bias, and choose to ignore it willfully. Want to learn about the best online casinos and the bonuses offered by them right now? Clicking here will take you to a website where you can compare the best online casinos of the modern times, what all they offer along with their current bonuses.
How to work out the Expected Value in a bet?
The Expected Value or EV tells us the extent to which we can expect to win (on average) on every bet placed by us, and is a pretty valuable calculation for every sports bettor, especially when he’s carrying out a comparison between different bookmakers’ odds. So, how do you think you can work out the Expected Value in your sports bets placed at some offline bookmaker or online casino, so as to predict your exact wins. Read on to discover.
EV or Expected Value is actually the amount you stand to lose or win if you were to place a wager several times, over and over again, with the same odds. For instance, let’s say you bet £ 10 on the possibility of tails in a coin toss at Bet365, and you’re promised £ 11 each time you get exactly that; the Expected Value in this type of bet would be 0.5.
It implies that if you place the same wager over and over again, you can expect to earn back at least £ 0.5 for every such £ 10 bet.
Expected Value or EV calculation
The formula used for calculating the Expected Value is quite easy – All you need to do is multiply the winning probability with the amount you stand to win on every bet placed by you, and then subtract that by the losing probability multiplied by the amount you stand to lose per bet:
EV or Expected Value = (winning probability x amount won per bet) – (losing probability x amount lost per bet)
When it comes to calculating the expected value in sports betting endeavours, you need to fill up the above formula with the decimal odds available to you. The process is as follows:
– Figure out the decimal odds for every outcome (draw, win or loss).
– Calculate the potential winnings for every outcome by multiplying the bet amount with decimal, thereafter subtracting the bet amount from it.
– Divide 1 by odds of a specific outcome, and calculate the probability/chances of that outcome.
– Apply this information to the formula provided above
So, for instance, if Chelsea (1.263) were playing Tottenham (13.500), and the odds of draw were 6.500 at an online casino such as Bet365, a £ 10 bet placed on the possibility of Tottenham winning the game, would yield possible returns of £ 125 from a win (the chances of that being 7.4% or 0.074).
The chances of that outcome not happening would be the sum of a draw and a Chelsea win, or in other words:
0.154 + 0.792 = 0.946
The loss incurred per bet is the initial bet amount – £ 10. Hence the complete formula would look something like as follows:
(0.074 x £ 125) – (0.946 x £ 10) = £ -0.20
As can be seen above, the Expected Value is a negative figure for this bet, implying that you stand to lose an average of £ 0.20 for every £ 10 bet you place.
How does the concept of Expected Value help in the field of sports betting?
If you recall, we had stressed that a negative Expected Value doesn’t imply that you are going to incur money loss on your bets. Unlike as in case of a coin toss, indulging in sports betting odds is quite subjective in nature, and hence your chances of making money are comparatively higher if you manage to outsmart the bookie.
You can easily see where exactly you can locate a positive Expected Value, and hence the best chances of winning, if you manage to calculate your own probability for a specific match which differs from the probability of the odds provided by the bookmakers.
To give you an example, as per the betting odds the chances of Tottenham scoring a win are pretty low at 7.4%. However, if you use a system such as Poisson Distribution and calculate the Tottenham’s winning chances to be 10% instead, the Expected Value for bets placed on the Tottenham win would jump to £ 3.262.
This is also the perfect means of comparing odds when you’re indulging in arbitrage betting.
Calculating the Expected Value of bets provides the sports bettors with detailed information about the value provided by the bookmakers. Although low margin bookmakers such as Bet365 may provide you Expected Value in the vicinity of £ -0.20, it isn’t uncommon for other regular bookmakers to provide you an Expected Value of around £ - 1.00 for every £ 10 bet placed by you (making you a probable loss of £ 1 per bet).