Odds and Probabilities, The Monty Hall Problem and The Hole-In-One Gang

Recognising Value – the Key to Gaining a Betting Edge. Read about the Monty Hall problem and the Hole in 1 gang.

Risk is part of everyday life, more so than most people probably realize. From crossing the road to the more obvious financial decisions such as buying a house, or starting a business, all involve varying amounts of uncertainty which must be considered. Gambling is the purest expression of risk, yet even when presented with a seemingly simple choice of potential outcomes for an unknown event, such as a football match, many bettors display a worrying ignorance of the concept of value and the fundamental mathematical principals involved. In simple terms, if a bettor cannot recognize ‘value’ they will never be a long term winner.

The Monty Hall Problem

Take a look at this seemingly simple mathematical puzzle, known as the Monty Hall paradox (named after the host of “Let’s Make a Deal”, a popular US show in the 60’s & 70’s which formed the basis of the poser).

An unbiased game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. “First you point toward a door,” he says. “Then I’ll open one of the other doors to reveal a goat. After I’ve shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door.” You begin by pointing to door number 1. The host shows you that door number 3 has a goat.

Do you gain value and see your chances of winning the car increase by switching to Door 2 or do you stay with Door 1 as it has an equal chance with only two doors left to choose from? When this question was posed in Parade magazine, 10,000 readers complained that the published answer was wrong – including several maths professors.

The assumption of “equal probability”, while being intuitively seductive, is wrong. The simple answer is to always switch doors. The car is behind one of the two closed doors, but you have no way of knowing which. Most contestants intuitively see no advantage in switching and assume that now there are only two doors, each must have an equal probability of revealing a car. In fact, your chances of winning the car actually double by switching to the door the host offers. If you switch, you gain value as theoretically you now have a 2/3 chance of winning the car. If you stayed with your original selection you have just a 1/3 chance of winning.

The principle is underlined by increasing the number of doors to 100. If 99 doors have a goat behind them and only one has a prize, if the player picks a door and then the host opens 98 of the other doors that were all shown to contain goats and then gives the player the opportunity to switch, the intelligent player would switch. The reason being that on average, in 99 out of 100 times the other door will contain the prize, as 99 out of 100 times the player first picked a door with a goat.

The Hole-In-One Gang

An excellent example of how this concept applies to betting was demonstrated by two sharp punters – Paul Simmons and John Carter – the self-styled Hole-In-One-Gang. In the summer of 1991, after studying the form, they calculated the chances of any given golfer in a tournament hitting a hole-in-one at around 50%. So they toured the UK placing maximum bets on the chances of a hole-in-one being scored by any player at a major that year. Lazy bookmakers who didn’t take the time to study the statistical likelihood put a finger in the air, and quoted amazing odds with 100-1 not uncommon.

That year, there were hole-in-one’s scored at 3 of the 4 majors and the pair’s winnings were reputed to be around £1million. Although it is difficult to put exact odds on a hole-in-one, it is clear that it is nowhere near 100/1. Due to the tradition of buying everyone in the clubhouse a drink after a successful hole-in-one, you can now buy insurance against it happening. Most insurers would probably refer to Francis Scheid’s (retired chairman of Boston University Maths Dept) 2000 study for Golf Digest. The magazine has kept hole-in-one stats since the 1950’s and Scheid put the odds of a Tour player scoring a hole-in-one at 3,000-1.

You can make a rough calculation for an average event like this week’s Johnnie Walker Championship at Gleneagles.


4 (short holes)*156 (players before cut)*2(rounds) PLUS 4*70 (players after the cut) * 2
= (1,248+560) 1,808 attempts against an average frequency of 1 in 3,000.

Probability Yes:
1,808/3,000=0.6026 or a 60% chance of occurring with true odds of 1.66

Probability No:
1,192/3,000=0.3973 or a 39% chance of occurring with true odds of 2.52

The hole-in-one gang were getting exceptional value on their bets playing at odds of 100/1 when in reality the chance of a hole-in-one occurring using Scheid’s figures was no more than a 2/3 (1.666) shot at true odds.

Such notions are all too common mistakes in gambling when bettors and bookmakers frequently act against their best interests. It doesn’t matter if it’s a game show, playing the lottery or sports betting, understanding and finding value is the key to profit. Like the Monty Hall question, successful betting requires the skill to understand whether the odds offered on an event represent the statistical probability of that event occurring – if it doesn’t then you will have an edge and gain value.