Risk comes from not knowing what you are doing

Warren Buffett

Gambling and Investing are not cut from the same cloth and yet, most investors tend to approach an investing problem like a gambling problem using the same bias, prejudices and far-fetched “logic & conviction” that this time the coin will turn a heads and a tail. Infact movies like 21, The Hangover and Ocean’s Eleven have popularized the notion that you can take down the house on your lucky day or better still, you can devise a system that is your sure-shot way to success on the casino floor.

In this blog post, we shall examine the player’s odds in different casino games and have a look at the research that has been done around this.

# A Quick History on Gambling

Gambling dates back to a time when history, as we know it, was not even written. Excavations at sites of the Mesopotamian civilization reveal the use of a six-sided dice dating back to around 3000 BC. The gambling dens of ancient China were widespread and bets were placed on animals fighting. By the 10th century, games like lotto and dominoes (a precursor to Pai Gow) appeared in China. Playing cards in China can be seen in the 9th century, the Japanese gambling pastimes were tracked to the 14th century and the Persian game of As-Nas dates itself to the 17th century.

The first known casino, the Ridotto, opened in 1638 in Venice, Italy. It was closed down in 1774 as the city government felt it was impoverishing the locals. In the United States, gambling dens were known as saloons (yes, as in the Westerner movies). In the early 20th century, gambling in the United States was banned by state legislation and social reformers. It wasn’t until 1931 that gambling was legalized in the state of Nevada. In 1976, the state of New Jersey allowed gambling in Atlantic City.

Gambling centres have come up all over the world now but some are more famous than others. Here’s a list of the most popular ones –

- Monte Carlo in Monaco is a popular casino and tourist attraction for the rich & famous. The casino there is featured often in James Bond novels and films.
- Casinò di Campione is located in the Italian enclave of Campione d’Italia in Switzerland. Founded in 1917, it is today Europe’s largest casino and a most popular gambling destination besides Monte Carlo. Such is the income from the casino, that it manages the entire operation of Campione without the need for any other revenue or taxes.
- Macau is a former Portuguese colony and a special administrative region of China since 1999. Macau recently surpassed Las Vegas as the world’s largest gambling market. The Venetian Macao is the world’s largest casino.
- Singapore is an upcoming destination for visitors who have the urge to gamble. The Marina Bay Sands is the world’s most expensive standalone casino and among the world’s ten most expensive buildings.
- The United States has over a thousand casinos. This number continues to grow as more states seek to legalize casinos. On last count, 40 states have some form of casino gambling. Las Vegas in Nevada has the largest concentration of casinos in the United States followed by Atlantic City, New Jersey and then Chicago.

# Casino Games

There are many types of casino games which use pure luck or a combination of luck and skill. The games are created as card games, slot games, table games, number games etc. Here’s a list of all popular casino games –

- Roulette
- Poker
- Blackjack (21)
- Baccarat
- Craps
- Slots
- Texas hold ‘em
- Keno
- Bingo
- Wheels of Fortune (The Big Six)
- Pai gow poker

Of these the top five favourite games among casino visitors are Slots, Blackjack, Roulette, Poker and Craps.

# What is House Edge?

A Casino is also named as the “House”. Casino games are designed to provide predictable, repeatable long-term advantage to the casino (or house) and yet, offer the players the general possibility of short to long-term gains in the short run. Yet, the more the player plays, the higher is the probability of the the house winning.

A house edge (or house advantage) can be created with the casino not paying the winning wagers according to the game’s true odds. For example – let’s play a game which’ll have the roll of one die. You can choose a number from 1 to 6 and if the roll happens to fall on your chosen number, you shall be paid four times of the money you have waged. Now, the true odds would be 5 times the amount wagered as there is a 1 in 6 chance of your winning number to show up. By paying only four times, the house has created an advantage for itself and a disadvantage for the player.

The house edge is defined as the casino’s profit as a percentage of the player’s original bet. Going back to our previous scenario, we see –

- Probability of winning = 1 on 6 (16.66%)
- Winning = 4
- Probability of losing = 5 on 5 (83.33%)
- Losing = -1

Therefore, the player’s expected value = [16.66% * 4] + [83.33% * -1] = 66.64% – 83.33% = -16.69%

In other words, the house edge is 16.69%

Let’s take another example.

In American Roulette, there are two zeroes and 36 non-zero numbers (i.e. 18 reds and 18 blacks). This gives the house a higher edge as compared to European Roulette which have only one zero.

So, in the American Roulette, the chances of a player who bets on the red color winning is 18/38 and the chance of him losing is 20/38. Thus the player’s expected value is calculated as – [18/38 * 1] + [20/38 * -1] = 0.473 – 0.526 = -5.26%

Thus the house edge (or house advantage) is 5.26%

On the other hand, the player’s expected value on a European Roulette table comes to –

[18/37 * 1] + [19/37 * -1] = 0.486 – 0.513 = -2.70%.

The European Roulette table’s house edge is lower at 2.70% making this version a more player friendly version as compared to American Roulette

The house edge of casino games varies greatly with the game.

Now the calculation of the house advantage for the roulette table was very easy however it is not the case for other games which require computer simulation. Further, in games like Blackjack where there are skills involved, the house edge is arrived on an optimal play basis i.e. without the use of advanced techniques such as card counting. Good blackjack have house edges below 0.5%.

Some games have an edge as low as 0.3% while for some games like Keno, the house edge goes up to 25%. Slots (a novice favourite) have a house edge of upto 15%.

## Lessons from the House Edge for Investing

### Identify predictable long-term advantages to win every time

Just like a casino, the world of investing also offers such long-term advantages if you look at it carefully. For example – Jeff Bezos from Amazon.com once aptly said

*I very frequently get the question: ‘What’s going to change in the next 10 years?’ .. I almost never get the question: ‘What’s not going to change in the next 10 years?’ And I submit to you that that second question is actually the more important of the two — because you can build a business strategy around the things that are stable in time … in our retail business, we know that customers want low prices, and I know that’s going to be true 10 years from now. They want fast delivery; they want vast selection. It’s impossible to imagine a future 10 years from now where a customer comes up and says, ‘Jeff I love Amazon; I just wish the prices were a little higher,’ [or] ‘I love Amazon; I just wish you’d deliver a little more slowly.’ Impossible. And so the effort we put into those things, spinning those things up, we know the energy we put into it today will still be paying off dividends for our customers 10 years from now. When you have something that you know is true, even over the long term, you can afford to put a lot of energy into it.”*

A smart investor will be able to identify this “house edge” on what the consumer will continue to need decades from now which also holds true today. This could mean better quality products, lower prices, beautifully crafted aesthetic products, money back guarantee, fast delivery, excellent after-sale services, prompt customer support, amazing deals, better returns, lower portfolio risk, wide selection, environmentally conscious, superior brand etc.

# What is Gambler’s Fallacy?

The gambler’s fallacy is also known as the Monte Carlo fallacy. It is a belief that if something happens more frequently (i.e. more often than the average) during a given period, it is less likely to happen in the future (and vice versa).

So, if the great Indian batsman, Virat Kohli were to score scores of 100 plus in all matches leading upto the final – the gambler’s fallacy makes one believe that he is more likely to fail in the final. Cricket commentators have a fancy phrase for it – “law of averages”.

In reality, in situations where the outcome is random or independent of previous trials, this belief turns out to be entirely false. So what Virat Kohli scores in the final has no bearing on his scores in the matches leading up to the big day. This fallacy arises in many other situations but all the more in gambling.

## The Monte Carlo Fallacy

The term is more popularly known as the “Monte Carlo fallacy” because of the events that took place in the Monte Carlo Casino on August 18, 1913.

The event happened on the roulette table where one of the gamblers noticed that the ball had fallen on black for a number of continuous instances. This got people interested and the “gambler’s fallacy” kicked in with the thought that since the ball had fallen on black so many times, it is likely to fall on red sometime soon. To derive “undue advantage of this certain-to-happen” belief, casino patrons started pushing money onto the table.

Well, the ball did fall on a red. But not until 26 spins of the wheel where each spin saw a greater number of people pushing their chips over to red. While the people who put money on the 27th spin won money, a lot more people lost their money due to the long streak of blacks.

## Everyday Examples of the Gambler’s Fallacy

While studies have found that asylum judges, loan officers, baseball umpires and lotto players employ the gambler’s fallacy consistently in their decision-making, all of us have at some time held the belief that a streak has to come to an end. We see this most prominently in sports where you predict that the 4th shot in a penalty tie-breaker is going to be saved because the last 3 went in.

Now we all know that the first, second or third penalty has no bearing on the fourth penalty (minus a little bit of pressure or release of it) and yet the fallacy kicks in. Similarly, if it hasn’t rained for 4 days during the rainy season, you tend to up the probability of rains pouring on the fifth day even though there is no scientific evidence to suggest so. We just can’t help thinking about the past in making future decisions even if there is no continuity in the process.

The gambler’s fallacy can also be illustrated with the repeated toss of a fair coin.

Now, the outcomes of a single toss are independent and the probability of getting a heads on the next toss is as much as getting a tails i.e. ½ for heads and ½ for tails.

Let’s toss the coin again and like the first toss, this one also lands on heads.

The probability of two heads in two coin toss is ½ x ½ = ¼ (i.e. 25% probability)

This is where the amateur investor starts to falter. He tends to believe that the chance of a third heads on another toss is still lower probability of ½ x ½ x ½ = ⅛ (12.5% probability). For this might be true if one were examining the probability of three heads before the start of the series. However, one has to account for the first and second toss to have already happened. This means the probability of a heads or a tails on the third toss is still ½ (and not ⅛ for heads and ⅞ for tails as the gambler’s fallacy might lead to)

We run this back into the Monte Carlo Casino game of 1913 and find the probabilities that gamblers might have assumed versus the real probabilities. Here is how the gambler’s fallacy plays –

- Spin 1 : There is a 50% probability of the ball landing on Black
- Spin 2 : There is 25% (50% x 50%) probability of the ball landing on Black
- Spin 3 : 12.5% probability (50% x 50% x 50%)
- .. and so on ..
- Spin 25 : 0.0000030%
- Spin 26 : 0.0000015% or “15 in a billion” or “1 in 66 million” probability
- Spin 27 : 0.0000007% or “7 in a billion” or “1 in 133 million” probability

Spin Number | The Fallacy (Assumed probability by gamblers of next spin coming as "Black") | Actual probability of next spin coming as "Black" |

1 | 50% | 50% |

2 | 25% | 50% |

3 | 12.5% | 50% |

4 | 6.25% | 50% |

5 | 3.125% | 50% |

6 | 1.5625% | 50% |

7 | 0.78125% | 50% |

8 | 0.390625% | 50% |

9 | 0.1953125% | 50% |

10 | 0.0976563% | 50% |

11 | 0.0488281% | 50% |

12 | 0.0244141% | 50% |

13 | 0.0122070% | 50% |

14 | 0.0061035% | 50% |

15 | 0.0030518% | 50% |

16 | 0.0015259% | 50% |

17 | 0.0007629% | 50% |

18 | 0.0003815% | 50% |

19 | 0.0001907% | 50% |

20 | 0.0000954% | 50% |

21 | 0.0000477% | 50% |

22 | 0.0000238% | 50% |

23 | 0.0000119% | 50% |

24 | 0.0000060% | 50% |

25 | 0.0000030% | 50% |

26 | 0.0000015% | 50% |

27 | 0.0000007% | 50% |

When the gamblers were done with Spin 25, they must have wondered – surely now with a 30 in a billion probability for it to come black, there is no way that the 26th spin will not land the ball in red. However that too did not happen.

Statistically, this thinking was flawed because the question was not if the next-spin-in-a-series-of-26-spins will fall on a red. The correct thinking should have been that in the next spin too, there is a 50:50 chance of the ball falling on a black or red square.

Numerous experiments have been done on the gambler’s fallacy which seems to lend more weight to the idea. A study by Fischbein and Schnarch in 1997 administered a questionnaire to five student groups from grades 5, 7, 9, 11, and college students. None of the participants had received any prior education regarding probability. The question asked was – “Ronni flipped a coin three times and in all cases heads came up. Ronni intends to flip the coin again. What is the chance of getting heads the fourth time?”

Now, we know that the answer to that is 50%, but here is how the students answered –

- Grade 5 – 35% times it will come heads
- Grade 7 – 35%
- Grade 9 – 20%
- Grade 11 – 10%
- College students – 0%

## Inverse Gambler’s Fallacy

The Inverse Gambler’s Fallacy is where after a series of events of a similar kind, the gambler believes that the series is bound to continue and is the more likely outcome. In our coin toss example, the gambler might see a streak of heads as a precursor to what is likely to come next – another heads. This too is a fallacy where the gambler presumes that the next coin toss carries a memory of past results which will have a bearing on the future outcomes.

Ian Hacking has described the inverse gambler’s fallacy as a situation where the gambler entering the room sees a person rolling a double six erroneously concludes that the person must be rolling the dice for some time as he feels it is very unlikely for someone to get a double six in their first attempt. Now, we know the probability of getting a double six is low irrespective of whether it is the first or the hundredth attempt. The fallacy here is the incorrect belief that the player has been rolling dice for some time.

## Retrospective Gambler’s Fallacy

The retrospective gambler’s fallacy is a situation where the gambler observes multiple successive “heads” on a coin toss and concludes that the previous unknown flip would have been a “tails”.

In his 1796 work “A Philosophical Essay on Probabilities”, Pierre-Simon Laplace wrote on the ways in which men calculate the probability of having sons. While the chances of having a boy or a girl child is pretty much the same, these men judged that if they have a boys already born to them, the more probable next child will be a girl. The expectant fathers feared that if more sons were born in the surrounding community, then they themselves would be more likely to have a daughter. It’s as if saying, “I am due the other” – we see this fallacy in many expecting parents who after having multiple children of the same sex believe that they are due having a child of the opposite sex.

## Scenarios where the Gambler’s fallacy does not apply

The gambler’s fallacy does not apply in scenarios where the probabilities of different events are not independent or mutually exclusive.

For example – in a deck of cards, if you draw the first card as the King of Spades and do not put back this card in the deck, the probability of the next card being a King is not the same as a Queen being drawn. The probability of the next card being a King is 3 out of 51 (5.88% probability) while that of it being a Queen is 4 out of 51 (7.84%). This effect is particularly used in card counting systems like in blackjack.

## Lessons from the Gambler’s Fallacy on Investing

### 1. Selective reporting

Statistics are often used to make content more impressive and herein lies the problem. When a public speaker says “the GDP is down this quarter”, it is safe to assume that the GDP was up the previous quarter.

This same problem persists in investing where amateur investors look at the most recent reported data and conclude on investing decisions. Like, reporting that the crude prices are forecasted to go up by 20% without mentioning that the prices were beaten down by 60% over the rest of the year.

### 2. Assuming small samples are representative of the larger population

The gambler’s fallacy arises from the belief that a small sample represents the larger whole. Studies by Daniel Kahneman and Amos Tversky have come to interpret that people believe short sequences of random events should be representative of longer ones. This means if you were to see a bunch of reds at point x and after a few randomness, you see another red streak – one tends to believe that the population is largely red with some small streaks of black thrown into the mix.

Investing entirely works on this. Often we see investing made on the premise that you can buy anything you want and because the macro-economic picture of the country is on a high, your stock will also go up. This is far away from the truth with a number of stocks currently lingering at their 52-week high even as the Indian Nifty and Sensex continues to touch new heights of 12,000 points and 40,000 points respectively.

### 3. The outcome is a result of the gambler’s skill

At some point in time, you would have had a streak of six when rolling dice. Notice how in your next roll, you will turn your body as if to have figured out the exact movement of the body, hand, speed, distance and revolutions you require to get another six on the roll. This is the gambler’s fallacy where the gambler believes that the outcome is the result of one’s own skill – a mistaken belief called the internal locus of control.

## How to Avoid the Gambler’s Fallacy?

### 1. View every event as a beginning

The best way to avoid the gambler’s fallacy is by treating each event as if it is a beginning and not continuation of previous events. This would prevent people from gambling when they are losing and avoid the mistaken-thinking that their chances of winning increase in the next hand as they have been losing in the previous events.

We see this in investing aswell where investors purchase stocks and mutual funds which have been beaten down on the hope that these would again rise up to their former glories which would result in a windfall for the investors. It is not uncommon to see fervent trading activity on stocks which are fallen angels or penny stocks.

### 2. Reduce your illusion of control of being able to predict events

The Gambler’s fallacy gets accelerated by an individual’s belief on one’s perceived ability to predict random events although it is not possible to predict these for truly random events. People who believe that have this ability to predict goes along the concept of them having an illusion of control.

This is very common in investing where investors taunt their stock-picking skills purely on the basis of being able to “feel” that a company is going to do very well or very bad in the near future. This is not entirely random as these stock pickers tend to offer loose arguments supporting their argument. You will do very well to not predict events without having adequate data to support your arguments.

# What is the Hot-Hand fallacy?

The hot-hand fallacy is typical of a game of basketball where people predict the same outcome as the previous event – known as positive recency – and conclude that a high scorer will continue to score. This is the exact opposite of the gambler’s fallacy where people predict the opposite outcome of the previous event or negative recency. Researcher like Ayton and Fischer have theorized that people display positive recency for the hot-hand fallacy because the fallacy deals with human performance. People believe that human performance is random and people are more likely to continue streaks which they believe is a process of continuous and hard practice over many years and honing one’s skills.

In investing, we see this behaviour is exhibited in over 90% of all trade that happens at the exchange and goes by the names of momentum (or swing) trading. The principle here is to bet on the stock that’s going up to go up further and vice-versa.

In one study, researchers gave a questionnaire to 455 participants ranging from age 22 to 90 years old. The two questions asked were – (1) Does a basketball player have a better chance of making a shot after having just made the last two or three shots than after having missed the last two or three shots? (2) Is it important to pass the ball to someone who has just made several shots in a row?

The main interest of the questionnaire was to see if a participant answered yes to the first question, implying that they believed in the hot-hand fallacy. The results showed that participants over 70 years of age were twice as likely to believe the fallacy than adults 40–49, confirming that older adults are more likely to remember positive information making them more sensitive to gains and less to losses than younger adults. The older participants seem to have more experience with seeing the hot-hand at play and often rely back on these experiences to get their answer. What the mind misses out is on the millions of moments when the hot-hand fallacy did not work out.

Investing too faces this bias as investors tend to remember hot-hands more intently and tend to get swayed by their previous experiences in determining investing choices.

*Sidenote: Recent research has found evidence supporting the hot-hand phenomenon. A paper by Yaari and Eisenmann examined a dataset of over 300,000 basketball free-throws and found the hot-hand at play. They found that there was a significant increase in players’ probabilities of hitting the second free-throw shot in a two-shot series if the first one went in as compared to if the first throw had not got in. Similar papers have been published by many researchers which lend to the idea that there exists a small yet statistically significant hot-hand phenomenon in sports where it comes to individual performance.*

# What is Gambler’s ruin?

The gambling game in the Mahabharata is well known to all Indians. The story goes like this –

*The Pandavas had arrived at Hastinapura, the capital city of the Kauravas, and were shown to the beautifully ornamented assembly hall. That is when Kauravas’s uncle Shakuni challenged Yudhisthira, the eldest of the Pandavas, to play a game of dice. Yudhisthira reluctantly agreed to the game and quickly started losing game after game to Shakuni, a past master in the art of gambling. Yudhisthira lost his jewels, his gold, his silver, his army, his chariots, his horses, his slaves and his kingdom. When Yudhisthira had lost every material possession, he put up his four brothers, his wife and himself up for wager and lost those aswell.*

The Gambler’s Ruin can be described in a number of ways –

1. A persistent gambler who raises his bet by a fixed amount when he wins, however does not reduce the amount wagered when he loses. Eventually, the gambler goes broke.

2. A persistent gambler with finite wealth will eventually go broke against an opponent with infinite wealth.

# What is Gambler’s Conceit?

The Gambler’s Conceit is a fallacy where a gambler believes he will be able to avoid (stop) a risky behaviour while engaging in it. In other words, the gambler believes that he will be able to exert self-control and stop playing while he is in positive cash territory. There is a phrase for this that goes by, “I’ll quit when I’m ahead.”

Quitting while ahead is unlikely because a gambler who is winning has little incentive or motivation to quit when luck in on his side. He is instead encouraged to continue to gamble away his winnings.

The gambler’s conceit frequently works in tandem with the gambler’s fallacy.

Casinos have a house advantage in games of chance and it is known that over the long run, the casino will take away all the player’s money. Thus, it is in the casino’s interest to keep a player on the tables for as long as possible thus lending support to the gambler’s conceit. Casinos offer players free alcoholic drinks to encourage them to keep gambling.

# What is Law of Averages?

The law of averages is the misdirected belief that a particular outcome is bound to happen eventually as long as it is statistically possible. It also adds that if an event at one end of the spectrum has happened (e.g. a team scores 500 runs in a game of 50-over cricket), sooner or later there will be a disastrous performance where a cricket team will be bundled for 50 runs in their innings. The logic provided here is that the law of averages will come into play.

The “law of averages” is nothing more than wishful thinking and displays a very immature understanding of statistics. A further corollary to the law of averages is the belief that if an outlier event (like 500 run inning) has happened, it is less likely to happen again in the near future to keep the fine balance fermented by the law of averages concept.

Let’s take an example. On the roulette wheel, you get reds on three consecutive spins. An onlooker uses this information and processes that the next spin is more likely to land on a black although the wheel has no memory of the previous spins. The onlooker was misguided by the law of averages using which he calculated that the probability of a red or a black is closer to 50%* and hence with 3 reds in a trot, he reckoned the black has come in next to move the average from the current 100% to 50%

* The probability of a black or a red is not exactly 50% because the roulette wheel has 37 or 38 pockets. The European Roulette wheel has 37 pockets where the 37th pocket is a “0” (zero). And the American Roulette wheel has 38 pockets where the 37th pocket is a “0” and the 38th pocket is a “00”. These 0 and 00 pockets offer the necessary edge to the house. Thus, your probability of the ball landing on a red pocket is –

- European Roulette = 48.65%
- American Roulette = 47.37%

# Bonus: What is the probability of getting 50 heads and 50 tails from 100 coin tosses?

This question also has a high amount of bias hidden in it. Most people believe that if a coin was tossed a 100 times, it will lead to an equal number of heads and tails i.e. 50 each. However, it might come to surprise you that the probability of landing 50 heads and 50 tails (in whatever order) from 100 coin tosses is only 8%. Let’s see how we calculated that.

We start small. Let’s say 5 coin tosses.

This gives us 32 combinations :

**5H**– HHHHH**4H,1T**– HHHHT, HHHTH, HHTHH, HTHHH, THHHH**3H, 2T**– HHHTT, HHTTH, HHTHT, HTTHH, HTHTH, HTHHT, HTHHT, THHTH, THHHT, THTHH**2H, 3T**– HHTTT, HTTTH, HTTHT, HTHTT, TTTHH, TTHHT, TTHTH, THTTH, THTHT, THHTT**1H,4T**– HTTTT, THTTT, TTHTT, TTTTH, TTTHT**5T**– TTTTT

Now, your answer to the question – “What likely combination will you get on 5 coin tosses? Is more likely to be “3 Heads and 2 Tails” or “2 Heads and 3 Tails”. The probability of you being right is only 62.5% (i.e. 20 out of 32 combinations).

Let’s run this same experiment over 10 coin tosses.

Number of Heads | Number of Tails | Chance of Landing this Combination |

0 | 10 | 0.1% |

1 | 9 | 1.0% |

2 | 8 | 4.4% |

3 | 7 | 11.7% |

4 | 6 | 20.5% |

5 | 5 | 24.6% |

6 | 4 | 20.5% |

7 | 3 | 11.7% |

8 | 2 | 4.4% |

9 | 1 | 1.0% |

10 | 0 | 0.1% |

Again, the law of averages may point towards 5 heads and 5 tails, however the data shows that the probability of landing 5 heads and 5 tails is just 24.6%.

The mid-point prediction turns even worse when we increase the number of tosses to 100

Number of Heads | Number of Tails | Chance of Landing this Combination |

40 | 60 | 1.1% |

41 | 59 | 1.6% |

42 | 58 | 2.2% |

43 | 57 | 3.0% |

44 | 56 | 3.9% |

45 | 55 | 4.8% |

46 | 54 | 5.8% |

47 | 53 | 6.7% |

48 | 52 | 7.4% |

49 | 51 | 7.8% |

50 | 50 | 8.0% |

51 | 49 | 7.8% |

52 | 48 | 7.4% |

53 | 47 | 6.7% |

54 | 46 | 5.8% |

55 | 45 | 4.8% |

56 | 44 | 3.9% |

57 | 43 | 3.0% |

58 | 42 | 2.2% |

59 | 41 | 1.6% |

60 | 40 | 1.1% |

The chance of getting 50 heads and 50 tails is now only 8.0%

In these cases, flexibility will be required in assessing the odds. For example – the chance of getting between 45 to 55 heads is 72.9% while that of getting between 40 to 60 heads is a high 96.5%.