Conjunction Fallacy and Security in Sports activities betting

Arnold Clark Cup

Overloading a betting ticket with a number of ‘certain bets’ for the sake of ‘security’ is an unwise technique in sports activities betting. The bettor may overestimate the possibilities of these occasions occurring collectively and as such, commit a conjunction fallacy when putting accumulator bets.

In sports activities betting there are two fundamental common methods that bettors could embrace:

One is that of chasing shock (which presents excessive payout odds) and putting a wager on such occasion, as a person wager or mixed with a number of different bets.

The opposite is that of betting solely on “nearly certain” occasions (such because the victory of the highest/robust groups), as a type of “security”.

The latter technique fits these bettors on the lookout for common small income fairly than low-chance large hits, because the payout odds of such “certain” bets are low (1.1 – 1.5 to 1 as a median vary). Nevertheless, the effectiveness of such a “secure” or “defensive” technique relies upon primarily on a subjective issue, that each bettor has to resolve on earlier than putting a wager, particularly what number of occasions they may select for an accumulator wager.

If security additionally assumes a low funding for the bettor, given the low payout odds of a “secure” wager, this may also indicate a low attainable revenue. The one solution to enhance the attainable revenue is by making a mixed wager with a number of such occasions, often known as an accumulator or parlay wager.

On this article we will focus on such “security”, see how the variety of occasions on the ticket could have an effect on it, and draw conclusions about what this quantity ought to be for preserving the wager comparatively secure.

Overloading the ticket in sports activities betting

Underneath the constraint of a set stake as funding and low payout odds, it’s regular for a bettor to have the impulse of loading their ticket with a number of bets. These sort of accumulator bets are pushed on the one hand, by the purpose of getting the next payout than for one wager and, however by the idea that the probabilities for a win don’t lower vastly if betting on a number of “certain” occasions fairly than a single occasion.

Let’s take a concrete instance.

Think about that the bettor chooses some matches the place the prior info signifies a favourite to win (as such, their payout odds might be low) and resolve to put an accumulator wager on these matches, by betting on the victory of the favourite groups.

Assume the bettor selected to wager on solely 4 such matches on a ticket, every having 1.2 as payout odds:

Match 1: victory of the favourite workforce; ×1.2 payout odds

Match 2: victory of the favourite workforce; ×1.2 payout odds

Match 3: victory of the favourite workforce; ×1.2 payout odds

Match 4: victory of the favourite workforce; ×1.2 payout odds

Complete (mixed) odds: 1.2 × 1.2 × 1.2 × 1.2 = 2.07

Assume the bettor estimates (subjectively) a chance of 70% for any of these “secure” occasions, assumed impartial of one another. Then the chance of their conjunction (the occasion that happens if every of these 4 occasions happens) is (70/100) × (70/100) × (70/100) × (70/100) = 24.1%. The probabilities decreased about 3 times in relation to the probabilities for a  single match, and the lower could be even greater with new matches added on that ticket. Chance is a subunitary quantity, and the product of any multiplication of chances would lower with the variety of components. Therefore, the “security” induced by a excessive chance for one match is drastically decreased when rising the variety of matches on the ticket.

How chance decreases with giant conjunctions of occasions

We noticed within the earlier part a concrete instance of how chance decreases with multiplication, as being subunitary. To grasp and understand the explanation of this lower on the whole, we’ve got to investigate it in its combinatorial nature. This lower is the impact of the facility of multiplication of combos and might be illustrated by way of unfolding. Within the following incomplete diagram, the attainable results of every match (1 – victory of the hosts, X – draw, and a couple of – victory of the visitors) is mixed with the attainable outcomes of the opposite matches as an unfolding of paths (every node stands for a match):

In whole, there are 3×3×3×3 = 81 paths and just one out of them, say 1–1–1–1, corresponds to a great prediction for the 4 matches. In a single single match, the nice prediction could be one out of three. After all, this diagram doesn’t replicate the actual possibilities, because the three attainable outcomes will not be equally possible – the truth is, the bettor assigned a 70% chance for the outcome 1 and the home additionally estimates this outcome as extremely possible. If it have been, the probabilities for a great prediction of a single match (1/3) would lower to 1/81 within the mixed wager.

We will replicate the assigned possibilities within the diagram by including extra equally possible nodes and implicitly paths, in a sure proportion. As an example, assuming 70% possibilities for 1, 20% for X, and 10% for two, the (incomplete) diagram would appear like the one under, the place every path is assigned a ten% probability:

Counting now the variety of paths favorable for the outcome 1–1–1–1, we discover 7×7×7×7 = 2,401 out of a complete of 10×10×10×10 = 10,000 attainable paths. Doing the ratio, we get 24.01% because the chance of that outcome.

When confronted with the confirmed lower in possibilities, the bettor is likely to be requested why they like to not place a wager with equivalent payout odds (about 2 : 1) on a special single occasion as an alternative of that mixed wager, supplied that the probabilities estimated for that single wager are greater than these of 24.01%. If the bettor doesn’t have a transparent reply for that, they could overestimate the chance of the conjunction, influenced by the nice possibilities of the favourite groups in these matches.

This overestimation falls inside what psychologists name the conjunction fallacy.

Conjunction fallacy and the way it impacts sports activities betting

Roughly talking the conjunction fallacy is an overestimation of the chance of a conjunction of occasions. If two occasions are impartial to one another, the chance of their conjunction is the product of their particular person chances: P(A and B) = P(A) x P(B). It follows immediately that the chance of the conjunction is lower than every particular person chance, since chances are subunitary numbers. A selected type of the conjunction fallacy as a mathematical error is when the topic takes the chance of the conjunction to be the sum, as an alternative of product, of the person chances. In its ordinary kind, the fallacy entails a false perception (based mostly on a sort of “feeling”) that the chance of the conjunction is greater than the chance of a person occasion throughout the conjunction. Cognitive psychologists suppose that the too-descriptive presentation of the measured occasions favours the fallacy.

Conjunction fallacy manifests in playing, particularly in these contexts that contain descriptive references of the measured occasions or through which the participant accesses prior info related to these occasions. Such circumstances are normally present in sports activities betting, and this is the reason conjunction fallacy is extra frequent in such video games than in others.

For instance:

Think about a horse race the place Horse 1 is the massive favourite (say it gained 5 out of the final seven races), and Horse 2 is a median horse with promising efficiency within the final races. Assume a bettor considers the next occasions for putting a wager on:

Horse 1 wins;

Horse 2 is the second;

Horse 1 wins and Horse 2 is the second.

Assume the bettor is extra involved with the probabilities to win than with the payout odds for the three occasions. If the bettor believes that occasion 3 is extra possible than one or each of the occasions 1 and a couple of (and accordingly he locations the mixed wager 1 and a couple of on the identical ticket), the bettor is topic to conjunction fallacy, since occasion 3 is a conjunction of the opposite two occasions.  This fallacy could flip right into a loss for the bettor, as they may wager on occasions 1 and a couple of on separate tickets, with greater possibilities every. However, they may win with that mixed wager as nicely. The idea is; that the idea the betting resolution was based mostly on was incorrect.

One other sort of occasion of conjunction fallacy in sports activities betting is the next:

In contemplating the bets related to a match between groups A and B, the bettor evaluates the possibilities of the next bets:

Group A scores one purpose;

Group B scores one purpose;

The ultimate rating of the match is 1 – 1.

If estimating the possibilities of wager 3 as greater than both of these of bets 1 or 2, the bettor commits a conjunction fallacy, since occasion 3 is the conjunction of the opposite two. Once more, the prediction for 1 – 1 may come true, however that may not validate the mistaken cause based mostly on a comparability of possibilities.

We will affiliate our examples of overloading a ticket with “certain” occasions within the earlier sections with a ordinary type of conjunction fallacy. The idea that the profitable possibilities don’t change a lot with the variety of matches within the mixed wager is conjunction fallacy additionally.

Regardless of the physiological components favorizing conjunction fallacy, it’s well-known that in sports activities betting, gamers normally have an impulse to mix as many bets as attainable on the identical ticket, to a sure subjective restrict, because the payout odds enhance with multiplication. We already noticed within the earlier sections that following this impulse shouldn’t be smart, however how ought to we deal rationally with the “secure” mixed bets in what issues the variety of bets on the ticket?

Splitting or collapsing the mixed wager to maintain it secure

In addition to a attainable conjunction fallacy, when it comes to optimum play if security is anxious, it’s safer to separate a ticket with “certain” or “nearly certain” occasions and its stake. In our preliminary instance, taking part in two tickets – holding two matches every – as an alternative of 1 and halving the stake in two is safer. Assume S is the stake of the unique ticket with 4 matches and S/2 the stake of every decreased ticket:

Ticket A, stake S/2:

Match 1: victory of the favourite workforce; ×1.2 payout odds

Match 2: victory of the favourite workforce; ×1.2 payout odds

Ticket B, stake S/2:

Match 3: victory of the favourite workforce; ×1.2 payout odds

Match 4: victory of the favourite workforce; ×1.2 payout odds

Every ticket has 1.44 : 1 payout odds. In case all 4 outcomes are predicted nicely, the 2 profitable tickets collectively give a revenue of 0.44S (much less the attainable ticket price charged by the company; normally on-line companies don’t cost such price), which, in comparison with the attainable revenue of the unique unsplit ticket, particularly 1.07S, is certainly decrease. Nevertheless, if one result’s predicted wrongly, one of many tickets A and B loses, and the general loss is (S/2) – 0.44×(S/2) = 0.28S, which is about 3 times decrease than the attainable loss with the preliminary ticket (S). If staying with the unique ticket, it will be shedding anyway on this scenario.

Subsequently, it’s safer to separate the ticket in two, which helps the concept that there shouldn’t be too many occasions on the identical ticket, nevertheless “secure.” After all, if the attainable revenue of the cut up ticket shouldn’t be passable for the participant, they’ll enhance its stake, at the price of the next attainable loss.

The query arises as what could be a security restrict for the variety of “certain” occasions on a ticket. There may be not an absolute reply to this query, because it relies on all people’s threshold of afforded danger and private betting technique. For evaluating the chance of any shock occurring for the predictions of such a mixed wager the one measure accessible is the general payout odds, that’s, the product of the payout odds of the constituent bets. A 3-option wager (as it’s that on ultimate outcome 1X2 in matches) with payout odds between 2 – 3 for one choice is normally categorized by the betting companies as random, that’s, any result’s attainable with about the identical probability. Subsequently, 2 as general payout odds for a mixed wager might be pretty chosen as a threshold for security. This selection can be supported by the choice of collapsing a mixed wager into a special particular person wager with the identical payout odds, for a similar stake: As a substitute of betting on a conjunction of a number of occasions, one could select to wager the identical quantity on a single occasion with the identical payout odds (because the product of the payout odds of the initially chosen mixed wager).

In our instance, as an alternative of betting on the victory of the 4 favourite groups (with 1.2 payout odds every), one could select a three-option particular person wager of any type with about 2 : 1 payout odds. By way of home’s evaluation, the 2 bets are comparatively equally prone to be gained. When you select the person wager because of private evaluation or utilizing some goal info, the selection will get extra justified. However even within the absence of such private re-assessment of the probabilities for the person wager, there’s nonetheless an argument that inclines the stability towards the person wager: Home’s evaluation is in fact topic to error and approximation, regardless that it depends on statistical-mathematical algorithms (if it weren’t, there could be no surprises within the matches).

The probability of misestimating will increase with the variety of occasions thought-about and a misestimation for an “nearly certain” occasion means overestimation for the possibilities of the (on-paper) favourite workforce. This assertion is true additionally from the standpoint of the bettor – the extra matches they analyze for predicting their outcomes, the extra possible is that they do it wrongly in some unspecified time in the future. Doing it wrongly for just one occasion makes the mixed wager loosing.


Overloading the betting tickets with a number of occasions thought as “nearly certain”, for the sake of “security”, shouldn’t be a smart technique. Such a mixed wager may need first rate payout odds, however the bettor may overestimate the probabilities that these occasions will occur collectively and as such, commit a conjunction fallacy. The extra “nearly certain” occasions on the ticket, the decrease the possibility to win.

The bettor really chooses “security,” however enlarging the variety of bets on the ticket (with the aim of getting greater payout odds) really reduces that security. After all, this conduct is within the benefit of the sports activities betting home.

A comparatively secure technique when betting on apparently “nearly certain” occasions is to restrict their quantity to have a most of about 2 general payout odds, splitting the meant mixed wager on this respect, or selecting a person wager with the identical payout odds for a similar stake as an alternative. After all, the arguments don’t apply for these bettors whose methods are based mostly on chasing surprises and the best attainable payout odds.


Barboianu, C. (2022). Understanding Your Sport: A Mathematician’s Recommendation for Rational and Secure Playing. PhilScience Press.

Nilsson, H., & Andersson, P. (2010). Making the seemingly unattainable seem attainable: Results of conjunction fallacies in evaluations of bets on soccer video games. Journal of Financial Psychology, 31(2), 172-180.

Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in chance judgment. Psychological assessment, 90(4), 293.

Author: Frank Campbell