Australia Sports Betting

In an earlier post I mentioned the fact that I had placed a bet on the Bulls in the semi finals of the Super 14 competition. The Bulls are highly fancied to win the tournament because they have home ground advantage and their stadium is at high altitude. This tends to sap the energy out of visiting teams who are used to sea level fixtures.

Oddly enough, bookmakers were offering 2.40 on the Bulls to win the tournament. This is generous because they were installed as heavy favourites in their semi final against the Crusaders. On top of that you knew that if they reached the final they would again be installed as the favourite. Based on the odds on offer for the Bulls to win the tournament compared to their odds to beat the Crusaders, I made the gamble that I could bet for and against the Bulls to win, and lock in a guaranteed profit.

Fortunately I was correct. Due to the predictability of the odds on offer, I was able to lock in a profit without any substantial risk. It was possible because the odds on the Bulls to win the tournament were too generous.

Note that draws aren’t possible for these fixtures because it is a knock out tournament.

On the 18th of May I placed $50 on the Bulls (the only South African team left) to win the tournament with the following odds.

To hedge my bets against the Crusaders in the semi finals I put $18 on the Crusaders to beat the Bulls at odds of 2.75. This would have left me with zero profit had the Crusaders won, but would have prevented a $50 loss.

The Bulls won the game which set up the following odds against the Chiefs in the final. Up to this point I have made a $50.00 bet on the Bulls and a $18.00 bet on the Crusaders, for a total of $68.00.

So how much should I bet on the Chiefs? I have two options. I could set up my bets so that I make a profit if the Bulls win, with no loss if the Chiefs win. Alternatively, I could set up the bets so that I make an equal profit regardless of who wins.

For the first option I would need to set an amount (x) that would cover all of my betting expenditure if the Chiefs win.

2.70x = 50 + 18 + x
1.70x = 68
x = $40

If the Chiefs win the final, I would secure a profit of ($40*2.70) – $18 – $50 = $0.00. If the Bulls win I would secure a profit of ($50*2.40) – $50 – $18 – $40 = $12.00.

By placing $40 on the Chiefs at the 2.70 odds, I would provide myself with a risk free profit regardless of the outcome. Because this hedge only makes money if the Bulls win, I refer to this as a biased arbitrage bet. You can learn more about arbitrage theory here.

If I wanted an equal payout regardless of who won the final, I would bet an amount (y) on the Chiefs such that:

Profit if Bulls win = profit if Chiefs win

($50*2.40) – $50 – $18 – y = 2.70y – $50 – $18 – y

$52 = 2.70y – $68

$120 = 2.70y

y = $44.44

Rounded to the nearest dollar, a $44 bet on the Chiefs would provide $6.80 if the Chiefs win, and $8.00 if the Bulls win.

Because I plan to watch the game live this weekend I have opted for the ‘unbiased’ option so that I can watch the match stress-free.

Always take a close look for arbitrage opportunities when a particular team is offered generous odds to win a tournament, but at the same time, is heavily favoured to win each individual game. By taking advantage of the 2.40 odds on the Bulls to win the tournament, I was able to bet against them in each fixture and secure a profit regardless of the result.

Anyone who has shopped around for casino betting strategies has likely heard of the Martingale betting system. It is a simple strategy where you double your bet size each time you lose, and reset your bet amount to a base level each time you win. The logic behind it is that you have to eventually win sometime, hence you will always return a profit. While this system is mathematically flawed for casino betting, I thought it would be interesting to apply it to sports betting. The fundamental difference between casino betting and sports betting is the real odds of winning aren’t fixed, and there are plenty of sports where long winning streaks are rare.

This article first discusses the Martingale betting strategy as it would apply to a roulette table at a casino. This is followed by Martingale inspired strategies for sport tournaments and other sport scenarios. I have come to the conclusion that if you pick your targets carefully, you can make the system work for you, but the system is by no means risk free, and it requires a substantial amount of funds in your betting account.

What is the Martingale betting strategy?

The martingale strategy can apply to a variety of games such as craps and coin tosses. It is explained below with respect to roulette.

The strategy works by setting a base bet amount, say, $10. You place a bet on red (or black, odds, or evens), which in the American roulette table has a 18/38 chance of winning and a 20/38 chance of losing (47% and 53%, respectively). If you win, you place another $10 bet on the next spin of the wheel. If you lose, you double your bet to $20 for the next spin. If you lose again, you bet $40, then $80, then $160, and so on, until you win. The idea is that streaks of black or green on the roulette table can’t go on forever, so you will have to win sometime, making this a ‘risk free’ way to earn money.

Why the Martingale strategy is flawed (when applied to casinos)

There are two primary flaws with the strategy. The first flaw is that most tables have a betting limit, so if you hit a nasty losing streak you could hit the table limit before you can win back your money. Some casinos, however, do offer no limit tables.

The second, and more fatal flaw, is that if you play the strategy long enough, you have a very good chance of running out of money before winning back your losses. This is because each time you lose on a spin, you increase your bet amount exponentially. The chance of losing six times in a row is (20/38)^6 = 2.1256%, which seems low, but the more spins you play, the odds of this happening increases dramatically.

- In 68 spins there is a 50.3% chance that you will lose 6 times in a row at some point
- In 150 spins there is a 80.6% chance that you will lose 6 times in a row at some point
- In 250 spins there is a 95.3% chance that you will lose 6 times in a row at some point
* Source: Wikipedia

This means you will need $640 left to avoid losing your money after six consecutive losses. Even if you bring $5,000 to the table there is a good chance you will reach six losses in a row on multiple occasions, and each time that happens you could only afford another two losses before running out of money.

You can also mathematically prove the system is flawed. Basically, it only work if the odds of winning are more than 50%, which never occurs in casinos with 1:1 returns. See Wikipedia for a mathematical proof.

Now that the Martingale strategy has been introduced, it will now be discussed with regards to sports betting.

The Martingale strategy applied to sports betting

Three Martingale type strategies will be looked at. The first strategy will be to bet against a player in a tennis tournament. The second strategy will be to bet against a team during the Major League Baseball (MLB) regular season. The third strategy will involve line bets.

Tennis tournament

In a tennis tournament there is typically only a small pool of players who have a reasonable chance of winning. In the current men’s game you would predict the winner of a grand slam to be either Nadal, Federer, Djokovic or Murray.

The Martingale strategy is to pick a player who you are convinced won’t win the tournament. You then consistently bet against this player until he or she is knocked out of the tournament. Each time you lose you will have to bet a larger amount to cover your losses from the previous bets. Because ever larger bets are required as they progress through the tournament, it is critical that you select your player carefully.

Tips for selecting a player

• Pick a player who you wouldn’t expect to get past the first round. You don’t want to still be betting against him or her in the quarter finals unless you have thousands of dollars in your account balance.
• Pick a player who is as far from the highest seeded players as possible. You want someone who will have to play lower seeded players before they reach a top eight opponent. This is critical to avoid a scenario where the odds on them losing is something like 1.08, which can often be the case against a top ranked player. Remember, the lower the odds, the more that must be spent to cover your losses from previous rounds.
• Pick a player that you know and who has a track record of not getting far. Avoid a wild card entry unless you know something about them. This is to avoid a scenario where a young prodigy comes in and reaches a late stage of the tournament.

Let’s say you pick Xavier Malisse to not win a tournament. In his first round he is drawn against James Blake with the following odds. You set your base bet at $10.00.

Malisse — 2.40
Blake — 1.55

Best case scenario

Blake beats Malisse and you win a ($10 x 1.55) – $10 = $5.50 profit. You can then focus on finding new targets.

Good case scenario

Malisse beats Blake and you lose $10.00. The next round’s odds for Malisse’s match against Juan Monaco are as follows.

Malisse — 2.80
Monaco — 1.40

To recoup your $10 in losses you will need to bet at least $10/(1.4 – 1) = $25 on Monaco.

You choose to bet $30 on Monaco and he wins. Your net profit for the tournament is ($30 x 1.40) – $27 – $10 = $5.00.

Bad case scenario

Malisee beats Blake and Monaco to setup an encounter against Juan Martin Del Potro. The odds are as follows:

Malisse — 3.30
Del Potro — 1.30

To recoup your $40 in combined losses you will need to bet at least $40/(1.3 – 1) = $133.33 on Monaco.

You choose to bet $140 on Monaco and he wins. Phew! Your net profit for the tournament is ($140 x 1.30) – $140 – $27 – $10 = $5.00.

Worst case scenario

Malisse wins the tournament! In this case you will have either run out of money sometime before then or you will have managed to lose more on him every round, without ever securing a win. This highlights why it is critical to select your target players carefully.

MLB regular season

In the United States they have sports like the NBA and MLB where the regular seasons seems to go on forever. For this reason it is typically a reasonable assumption that a weak team will lose at least one more game before the season’s end.

In the 2009 MLB season the Washington Nationals currently have a 11 and 28 record (11 wins, 28 losses). Last season they finished the year 59 and 102, which means there are about 122 games left in their 2009 regular season. It is safe to say they will lose at least one game during that time!

The Martingale strategy involves setting a base bet of, say $10.

Washington’s next game is against the Pittsburgh Pirates with the following odds:

Pittsburgh Pirates — 1.98
Washington Nationals — 1.83

If Washington loses you receive $9.80 in profit. You would then bet $10 on their next game. If Washington wins you will have to recoup that loss in their next game against the Baltimore Orioles. If the odds are:

Baltimore Orioles — 1.63
Washington Nationals — 2.20

You would have to bet at least $10/(1.63 – 1) = $16 on Baltimore to recoup your losses.

If you bet $20 on the next game and Baltimore won you would win a $20*1.63 – $20 – $10 = $2.60 profit. If Washington won again you would have to bet more again on the next bet, and so on.

With this strategy you can be reasonably confident that NO team in the MLB will go undefeated through the 100+ games left in the regular season. This means you could choose to target any team. Keep in mind that the lower ranked teams will have lower odds on them to lose, meaning you will have to bet larger amounts to recoup previous losses.

How much money is needed to make this system work?

This all depends on the nature of your targets and your selected sport.

The best way to work this out is to run a scenario analysis. You will need to calculate how long of a losing streak you wish to be capable of handling. You then work out a reasonable base bet amount from there. If you’re like me and only have a access to a few hundred dollars, you will likely start with a $1 or $2 dollar bet. Let’s use the baseball example and say that you are confident Washington will never go on a five game winning streak. With the following odds for them to lose (keeping in mind that the odds will drift higher as their streak develops):
Odds

Game 1 – 1.98 – Washington wins
Game 2 – 1.63 – Washington wins
Game 3 – 1.72 – Washington wins
Game 4 – 2.12 – Washington wins
Game 5 – 1.62 – Washington wins
Game 6 – 2.00

Bets (rounded up)

Game 1: $1.00
Game 2: $1 / (1.63 – 1) = $2.00
Game 3: ($2 + $1) / (1.72 – 1) = $6.00
Game 4: ($6 + $2 + $1) / (2.12 – 1) = $10.00
Game 5: ($10 + $6 + $2 + $1) / (1.62 – 1) = $32.00
Game 6: ($32 + $10 + $6 + $2 + $1) / (2.00 – 1) = $52.00

If Washington lost their sixth game you would secure a profit of ($52 x 2.00) – $52 – $32 – $10 – $6 – $2 – $1 = $21.00.

Your total bet amounts are $103.00, so in this scenario you needed access to more than 100 times your base bet in order to sustain the five game winning streak.

The calculated required amount heavily depends on expected odds. For example, had the odds for game six been 1.20 you would have needed $302.00 in your account. Because the season is so long, you could sit out a game if you don’t like the odds, and wait for better value.

I would recommend following a team closely to establish an expected length of future winning streaks. You could then factor in a safety margin and then calculate your optimal base bet and account balance accordingly.

Martingale strategy applied to line bets

Line bets offer equal payouts to each team, and are designed to have a 50% chance of winning. Because one team will have a higher expected chance of winning, line bets are created by adding a predetermined amount to the underdog’s score, or by subtracting that same amount from the favourite’s score.

For a Martingale strategy applied to line bets, let’s look at it mathematically for one round.

Let q be the probability of losing a line bet. We then define the probability of winning a line bet as (1-q). Let B equal the base level starting bet. Finally, let n equal the number of bets you can afford to lose before running out of money.

The probability of losing all of your bets and running out of money is qⁿ. If you lose all of your bets the amount you would lose is:

If you do not lose all n bets, you win B (your initial bet amount). The expected profit per round is

For the profit to be greater than zero, the following must hold:

Recall that q is the probability of losing a particular bet, and the probability of winning a bet is 1-q. Below is a table of the required probabilities of winning each bet to provide an expected profit that is greater than zero. Each win probability in the table provides an expected profit of zero for the given n and line bet odds. The results are provided for when the line bet odds are 1.91 and when they are 1.95. These represent two common bookmaker odds on line bets.

The more bets you are able to make, the greater the required probability of winning each bet. Also, the required performance is greater for 1.91 odds than for 1.95 odds. As n approaches infinity, the required win ratio equals the inverse of the odds (i.e. 1/1.95 = 0.5128 and 1/1.91 = 0.5236).

Basically, if you can win line bets more than 53% of the time, you can make this Martingale strategy work for you.

So is the Martingale strategy flawed for sports betting?

If you choose your bets VERY wisely, you can make this system work for you, but it is by no means full proof. You will need to have a substantial amount of funds in your account to make it work, as well as nerves of steel if you make a large run of losses.

Personally, I will give the Martingale strategy a miss.

Sources:
Wikipedia – Martingale betting system

It’s always a good idea to keep an eye out for irregularities in available odds, because bookmaker mistakes do happen.

Last night Sportsbet was offering the following odds on the Super 14 Tournament outright winner.

The Bulls are the only South African team left in the tournament, hence the disparity in the 2.40 South Africa odds to the 2.35 odds on the Bulls.

I checked the odds again today and found that the winning country odds are now as follows.

So the disparity has been fixed.

It just goes to show that disparities do exist, although they disappear quickly!

For the Super 14 tournament I have placed a bet a South African team to win using the 2.40 odds offered last night. Providing the Bulls manage to beat the Crusaders, I will likely hedge my bets by betting on the winner of the Chiefs / Hurricanes match to win the tournament. The Bulls would have home game advantage, and they play at altitude, so providing they can beat the Crusaders I would expect them to be installed as favourites to win the final. Odds of more than than 1.72 on their opponent would enable me to lock in a profit before the final commences. All that needs to happen now is for them to beat the Crusaders! Fingers crossed.

While you shouldn’t really isolate week by week performance, it does feel good to end a week with more money in your account than what you started with. It is also good to end in the black if you have placed bets using a substantial portion of your account balance. This article looks at the chances of breaking even for a given number of line bets.

What is a line bet?

 
Line bets are designed to provide equal payouts for each team for a given event. Because one team will have a higher expected chance of winning, line bets are created by adding a predetermined amount to the underdog’s score, or by subtracting that same amount from the favourite’s score. Take, for example, SportsBet odds on Monday’s NBA fixture between the Rockets and the Lakers.

The LA Lakers are heavily favoured to win, hence the low 1.07 odds. The line bet provides equal 1.91 odds by giving the Houston Rockets a 12 point head start. At the end of the game, 12 points are added to Houston’s score, and the line bet payouts are determined based on their revised score compared to the LA Laker’s score. Conversely, you can think of it as subtracting 12 from the LA Laker’s score and comparing it to Houston’s original score.

Update: In this case the LA Lakers won the game by 19 points, so the line bet on the Lakers paid out.

How many bets should I make?

 
Line bets are designed to give a 50% chance of winning, but due to bookmaker margins, the offered odds are typically between 1.91 and 1.95. This means you have to win more than 50% of your line bets to achieve a net profit.

I have often setup two line bets, only to realise that I will have to win both of them to earn a profit. By adding a third bet, I will only need to win two out of the three bets to be in the black. So, what number of line bets gives the greatest chance of making a profit?

The table below lists the number and percentage of bets required to earn a net profit from a series of line bets.

The first column is the number of line bets made. The second column shows the number of wins required to make an overall profit. The ‘Req %’ column lists the percentage of bets that must win to make a profit. The probability column shows the chance of winning at least that number of bets. The final column shows the ratio of this probability to the percentage of winning bets required. This provides a measure of value, with a higher percentage meaning you have better odds of breaking even.

For example, if you made seven line bets, you would need to win 57% of them. The chance of this occurring (providing the odds on the line bets really are 0.5) is 50%.

There are two trends in the required percentage. First, an odd number of bets requires a lower percentage of correct bets to be profitable than the even bets either side of it. Second, the higher the number of bets (comparing odd numbers to odd numbers, and evens to evens), the lowered the required win ratio.

For the probabilities of making the required number of bets, they are always 50% for an odd number of bets, and less than 50% for even numbers. The higher the even number of bets, the closer to 50% the odds become.

As you can see, it is the most difficult to break even when you make two line bets. If the chance of winning each bet really is 50%, you have a 25% chance of winning both bets, hence you have a 25% chance of securing a profit. This is much worse than the 50% chance if you only made one bet. If you made a third bet you would then only need to win two out of three bets to make a profit. The probability of this happening is 50%, so you have twice as good a chance of making money!

If you’re confused as to why you have a 50% chance of winning at least two out of three bets, consider the possible outcomes:
0 wins
1 win
2 wins
3 wins

There are four possible outcomes, half of which are ‘in the money’. In terms of comparative probabilities:
P(0) = P(3)
P(1) = P(2)

And P(0) + P(1) +P(2) + P(3) = 1

So P(2) + P(3) = P(0) + P(1). Hence you have a 50% chance of winning at least two out of three bets.

Caveats

 
There are two caveats to this theory. First, you shouldn’t isolate performance from week to week. For example, you may decide to make five line bets per week, but over a two week period, this amounts to 10, so you would have had a better chance betting on five in week one and six in week two (providing you were never going to make a line bet again). This theory applies when you only occasionally make line bets. For example if you see two line bets that interest you for a weekend of rugby league, the theory shows you would be better off picking a third line bet to go with them.

Second, this theory assumes you have an equal chance of winning each bet regardless of how many bets you make. In reality, you may have stronger opinions regarding some bets than others. So if you have two bets you feel strongly about, and are looking around for a third line bet to complement them, you may end up picking a bet that has a lower chance of winning than your first two bets.

Notes

 
* The probability of making a profit for a series of line bets is calculated as the odds of making each number of bets over and above what is required. For example, if you have made five line bets, three of those bets must win to make a profit. The probability of achieving this equals the sum of the probabilities of winning 3, 4, and 5 bets. To calculate the probability of winning a particular number of bets, I have used binomial distribution theory. To calculate the binomial coefficient, I have used Pascal’s triangle, although you can use the formula given below. See the links at the bottom of the article for more information.

Binomial probability mass function

Where n is the number of bets, k is the number of winning bets, and p = 0.5.

Binomial coefficient

Pascal’s triangle

Sources:
Wikipedia – Binomial distribution
Wikipedia – Pascal’s triangle

Last weekend I created two multibets using English Premier League matches.

Bet 1 – 8.31 odds

Manchester United to beat Middlesbrough
Arsenal to beat Portsmouth
Liverpool to beat Newcastle
Chelsea to beat Fulham
Aston Villa to beat Hull City

Bet 2 – 24.10 odds

Manchester United to beat Middlesbrough
Arsenal to beat Portsmouth
Liverpool to beat Newcastle
Chelsea to beat Fulham
Aston Villa to beat Hull City
Everton to beat Sunderland

I got every pick right, winning both multibets!. The win on the 24.10 odds bet is now my highest ever odds winning bet. Comparing a $6 multi bet vs 6 $1 individual bets, the multi vs. single bet payouts and profits for these games are as follows:
Multi – $144.60 payout and $138.60 profit
Single – $10.64 payout and $4.64 profit

Three cheers for multibets! To top it off, I had also placed individual bets on each of the above events, and I had placed a winning bet on Manchester City to beat Blackburn. Had I added this match to the multibets I would have won on 46.27 odds!

It’s funny how sometimes you can look at a weekend’s fixtures and feel strongly about a number of matches, while other weekends you don’t feel confident about making any bets. I can’t remember the last time I placed bets on seven out of ten premier league games. I’ve had a look at next weekend’s fixtures, and I will probably only make a couple of bets.

A good thing about multibets on fixtures that span a number of days is, if you’re sill on track to win halfway through the fixtures, you can hedge your bets before the final game(s) to lock in a guaranteed profit.