Gambling Information at Craps 7
House Edge
There are many definitions for the term "house edge", "house advantage" or "HA".
Some are confusing to the average player and seem to contradict themselves.
Understanding what house edge really means to a player of casino games is not as easy as is adding 1+1. Well, at least for most of us that have an "average" working knowledge of math. We will proceed to show you a few simple examples of how the house edge works on casino bets.
Simply put the house edge it is the difference (expressed as a percentage) between what the true odds pays and what the actual casino payoff odds pays divided by the true odds total payout. Exactly.
As with most math calculations there are a few ways to calculate exactly the house edge on any bet. Some calculations are more difficult than others.
American Roulette Example. A one number bet in roulette pays 35 to 1. The true odds pay 37 to 1. When you win you win a total of $36 including your bet. You should have won $38 including your bet. The difference is $2 between the two payoffs and the house edge would be 2/38 which is 5.26%.
The house edge does come into play every time a player wins a bet by the casino not paying a true odds payout.
Another way to calculate the house edge is by first figuring out the expected value of a particular wager.
The expected value is the
theoretical average of the most common outcome that can happen in a series of trials making the same bet. The term "expected value" can be misleading. It must not be confused with the "most probable value" or "most probable outcome". Here we will use just one trial or bet to show the relationship expected value has to the house edge.
Using the roulette example above:
The formula is Expected Value (EV) = Bet ($1)
times the payoff, (35) times probability of winning (1/38) PLUS (+) Loss (-$1) times probability of losing (37/38).
EV = ((B*Odds Payout)*p) + (-B*q)
EV = (($1*35)*1/38) + (-$1*37/38) or (0.921052)+(-0.973684)= -$0.052631 (5.26 cents) which is the expected LOSS for a $1 bet. Multiply 0.052631 by 100 to get percent of 5.26% which is the house edge.
Of course it is impossible to lose exactly 5.26 cents for a $1 bet, but remember the expected value is the most common outcome of a series of identical bets, so you need more than one bet to have the EV come close to the actual most common outcome.
Other acceptable meanings of house edge are:
The average percentage of a player's wager that the house earns as a result of making a less than true payoff is termed the house advantage.
The house edge is
also defined as the ratio of the average loss to the initial bet.
- House edge is the difference between what the true odds pays and what the actual casino payoff odds pays divided by the true odds total payout.
- The lower the house edge, the losses for the player will be lower and the wins for the player will be higher.
- The higher the house edge, the losses for the player will be higher and the wins for the player will be lower.
Example number 2
Every casino bet you make has a certain winning or losing probability. The probability of a bet can be calculated exactly with math.
Most casino bets have less than a 50% chance of winning.
Let's discuss a wager on the toss of a coin, (I know a casino does not have a bet on a coin toss) the probability of heads or tails is 50% or 50/100. We call this an even money bet. A 50 - 50 chance of winning.
If you bet $1 that the next coin flip was "heads" and you win, and the payoff is $1, you would have received a TRUE ODDS payoff. This is close to the situation about the "Odds Bets" in the game of Craps.
However if the bet only paid you 98 cents every time you won instead of a dollar the house edge would be
1%.
.02 ( 2 cents) / $2 (the total true payout including the bet) = 1%.
The house keeps the 2 cents it should have paid in order to be a true odds payout.
When you lose a bet, you lose the whole dollar and it now takes more wins than losses to show a profit. The house edge in this example is the difference between the true odds and the odds that the casino pays you when you win.
The expected loss of the above example is 2 cents. Of course it is impossible to lose 2 cents on each bet you make. We need to see the effect of the house edge on many wagers to get a true picture of how the house edge works.
- Example of a bet on a coin toss
- probability of heads or tails is 50% or 50/100
- a TRUE ODDS payoff would be $1 paid to $1 bet
- A 1% house edge would be from a $1 wager that only pays 98 cents when the bet wins
The house advantage or house edge is built into almost all casino
bets to guarantee that the casino will make a profit in the long run.
The casino relies on many players and wagers to accomplish this.
The house edge does NOT guarantee that all players will lose in the short run and some may even be ahead in the long run but the possibilities or probabilities of that happening by chance are very low and always less than a 50% chance of winning.
Why is it important to understand the house edge?
The lower the house edge, the losses for the player, when the player loses, will be lower and the wins for the
player, when the player wins, will be higher.
The higher the house edge, the losses for the player will be higher when the player loses and the wins for the player when the player does win will be lower.
| Outcomes | Payout→ | $1 | 99 cents | 98 cents | 97 cents | 95 cents | 90 cents | 80 cents | 70 cents | Wins |
|---|---|---|---|---|---|---|---|---|---|---|
| Wins | Prob | 0% HA | .5% HA | 1.0% HA | 1.5% HA | 2.5% HA | 5% HA | 10% HA | 15% HA | Prob |
| 0 | 0.10% | -$10.00 | -$10.00 | -$10.00 | -$10.00 | -$10.00 | -$10.00 | -$10.00 | -$10.00 | 0.10% |
| 1 | 0.98% | -$8.00 | -$8.01 | -$8.02 | -$8.03 | -$8.05 | -$8.10 | -$8.20 | -$8.30 | 0.98% |
| 2 | 4.39% | -$6.00 | -$6.02 | -$6.04 | -$6.06 | -$6.10 | -$6.20 | -$6.40 | -$6.60 | 4.39% |
| 3 | 11.72% | -$4.00 | -$4.03 | -$4.06 | -$4.09 | -$4.15 | -$4.30 | -$4.60 | -$4.90 | 11.72% |
| 4 | 20.51% | -$2.00 | -$2.04 | -$2.06 | -$2.12 | -$2.20 | -$2.40 | -$2.80 | -$3.20 | 20.51% |
| 5 | 24.61% | $0.00 | -$0.05 | -$0.10 | -$0.15 | -$0.25 | -$0.50 | -$1.00 | -$1.50 | 24.61% |
| 6 | 20.51% | $2.00 | $1.94 | $1.88 | $1.82 | $1.70 | $1.40 | $0.80 | $0.20 | 20.51% |
| 7 | 11.72% | $4.00 | $3.93 | $3.86 | $3.79 | $3.65 | $3.30 | $2.60 | $1.90 | 11.72% |
| 8 | 4.39% | $6.00 | $5.92 | $5.84 | $5.76 | $5.60 | $5.20 | $4.40 | $3.60 | 4.39% |
| 9 | 0.98% | $8.00 | $7.91 | $7.82 | $7.73 | $7.55 | $7.10 | $6.20 | $5.30 | 0.98% |
| 10 | 0.10% | $10.00 | $9.90 | $9.80 | $9.70 | $9.50 | $9.00 | $8.00 | $7.00 | 0.10% |
An example:
Let us say your first session in the above game of 10 coin tosses ends up in winning only 2 heads, you choose heads because you are a heads up person. You also have a 4.39% chance of that
occurring or a 1 in 22.7 of your sessions this is expected to happen. With a 0% house edge, you would show a $6.00 net loss.(2 win +$2, 8 Loss -$8 = net $6). The next session you play, it would be nice to get 8 wins and 2 losses. It has the same chance or probability of actually happening. And
when it does happen you net exactly $6. So after two sessions, same amount of wins and losses, you are even. A nice
position to be in.
One more session win and you have a net win for your 2 out of three sessions.
Same example, now let's use the .5% HA column:
Let us say your first session in the above game of 10 coin tosses ends up in winning only 2 heads, you choose heads because you are still a heads up person. You also have a 4.39% chance of that
occurring or a 1 in 22.7 of your sessions this is expected to happen. With a .5% house edge, you would show a $6.02 net loss. That is because when you did win, you were only paid 99 cents for your win, but when you lost , you still lost the $1.
Now, the next session you play, it would be nice to get 8 wins and 2 losses. It
has the same chance or probability of actually happening. And when it does
happen you net exactly $5.92. So after two sessions, same amount of wins and
losses, you are now "in the hole" for 10 cents. A different position to be in.
Now you have to win 2 out of the first 3 sessions to show a profit.
The 10 cent difference may not seem like much but it is how the casino is guaranteed to win as long as it has players gambling and the casino will win more the longer those players play because they pay less than the true odds of hat the bet should pay but they do keep all your losses.
From the players standpoint, now it takes more wins the higher the house edge is. And the longer you play or the longer your playing sessions are you have to be able to get more wins than losses to, in relation to the expected number of wins you can expect.
More examples coming soon, because this is an important factor in considering how one plays, the type of wagers that are made and your "system of play".
| trials | 10 | outcomes | 50 | outcomes | 100 | outcomes |
|---|---|---|---|---|---|---|
| losing | 37.695% | 5 | 44.386% | 25 | 46.021% | 50 |
| winning | 37.695% | 5 | 44.386% | 25 | 46.021% | 50 |
| break even | 24.609% | 1 | 11.228% | 1 | 7.959% | 1 |
| outcome W% | 45.45% | 49.02% | 49.50% | |||
| trials | 500 | outcomes | 1000 | outcomes | 10000 | outcomes |
| losing | 48.217% | 250 | 48.739% | 500 | 49.601% | 5000 |
| winning | 48.217% | 250 | 48.739% | 500 | 49.601% | 5000 |
| break even | 3.566% | 1 | 2.523% | 1 | 0.798% | 1 |
| outcome W% | 49.90% | 49.95% | 49.9950% |
| trials | 10 | outcomes | 50 | outcomes | 100 | outcomes |
|---|---|---|---|---|---|---|
| losing | 62.305% | 6 | 55.614% | 26 | 53.979% | 51 |
| winning | 37.695% | 4 | 44.386% | 24 | 46.021% | 49 |
| break even | 0.000% | 0 | 0.000% | 0 | 0.000% | 0 |
| outcome W% | 40.00% | 48.00% | 49.00% | |||
| trials | 500 | outcomes | 1000 | outcomes | 10000 | outcomes |
| losing | 55.335% | 252 | 56.281% | 503 | 69.497% | 5026 |
| winning | 44.665% | 248 | 43.719% | 497 | 30.503% | 4974 |
| break even | 0.000% | 0 | 0.000% | 0 | 0.000% | 0 |
| outcome W% | 49.60% | 49.70% | 49.7400% |
| trials | 10 | outcomes | 50 | outcomes | 100 | outcomes |
|---|---|---|---|---|---|---|
| losing | 62.305% | 6 | 66.409% | 27 | 69.135% | 53 |
| winning | 37.695% | 4 | 33.591% | 23 | 30.865% | 47 |
| break even | 0.000% | 0 | 0.000% | 0 | 0.000% | 0 |
| outcome W% | 40.00% | 46.00% | 47.00% | |||
| trials | 500 | outcomes | 1000 | outcomes | 10000 | outcomes |
| losing | 88.639% | 264 | 95.316% | 527 | 99.9999932224% | 5264 |
| winning | 11.361% | 236 | 4.684% | 473 | 0.000006778% | 4736 |
| break even | 0.000% | 0 | 0.000% | 0 | 0.000% | 0 |
| outcome W% | 47.20% | 47.30% | 47.3600% |
Lessons learned from the above charts are the longer one plays the less probability you have of winning. So keep your gambling sessions towards the short side.
Lesson Two is
to choose between bets that have lower house edges so you can have better chances at winning
the most $$$ when you do win and losing the least $$$ when you do lose.
The higher the house edge the higher the total
% of losing outcome possibilities and the lower the total % of winning outcome possibilities as more bets are made. The house edge can not and does not change the probability of any single outcome.
Look at the 3 popular casino bets in the table below. The 'Banker' bet in Baccarat has the lowest House Edge but has the HIGHEST Losing % but has the highest winning % also.
Coin toss charts and tables are easy to create and understand how
things relate to each other but you will not find a coin toss game in
any casino. It could be done but it makes you wonder why it has not yet
been attempted.
So, let us look at a few actual casino bets that one can find in a real casino.
| Wins | Prob | NET | Bet | Wins | Prob | NET | Bet | Wins | Prob | NET | Bet | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0001% | -$100.00 | Pass Line | 0 | 0.0003% | -$100.00 | RED | 0 | 0.0001% | -$100.00 | Banker | ||
| 1 | 0.0025% | -$90.00 | Craps | 1 | 0.0048% | -$90.00 | Roulette | 1 | 0.0015% | -$90.25 | Baccarat | ||
| 2 | 0.0227% | -$80.00 | 2 | 0.0409% | -$80.00 | 2 | 0.0146% | -$80.50 | |||||
| 3 | 0.1323% | -$70.00 | lose | 3 | 0.2211% | -$70.00 | lose | 3 | 0.0902% | -$70.75 | lose | ||
| 4 | 0.5464% | -$60.00 | 43.70% | 4 | 0.8457% | -$60.00 | 50.63% | 4 | 0.3937% | -$61.00 | 56.47% | ||
| 5 | 1.6998% | -$50.00 | win | 5 | 2.4355% | -$50.00 | win | 5 | 1.2935% | -$51.25 | win | ||
| 6 | 4.1310% | -$40.00 | 38.72% | 6 | 5.4798% | -$40.00 | 32.23% | 6 | 3.3202% | -$41.50 | 43.53% | ||
| 7 | 8.0315% | -$30.00 | even | 7 | 9.8637% | -$30.00 | even | 7 | 6.8180% | -$31.75 | even | ||
| 8 | 12.6872% | -$20.00 | 17.58% | 8 | 14.4256% | -$20.00 | 17.14% | 8 | 11.3757% | -$22.00 | 0.00% | ||
| 9 | 16.4445% | -$10.00 | 9 | 17.3108% | -$10.00 | 9 | 15.5734% | -$12.25 | |||||
| 10 | 17.5845% | $0.00 | 10 | 17.1377% | $0.00 | 10 | 17.5890% | -$2.50 | |||||
| 11 | 15.5401% | $10.00 | 11 | 14.0217% | $10.00 | 11 | 16.4178% | $7.25 | |||||
| 12 | 11.3300% | $20.00 | Expected | 12 | 9.4647% | $20.00 | Expected | 12 | 12.6428% | $17.00 | Expected | ||
| 13 | 6.7779% | $30.00 | Loss | 13 | 5.2420% | $30.00 | Loss | 13 | 7.9883% | $26.75 | Loss | ||
| 14 | 3.2944% | $40.00 | $1.41 | 14 | 2.3589% | $40.00 | $5.26 | 14 | 4.1010% | $36.50 | $1.06 | ||
| 15 | 1.2810% | $50.00 | HA | 15 | 0.8492% | $50.00 | HA | 15 | 1.6843% | $46.25 | HA | ||
| 16 | 0.3892% | $60.00 | 1.41% | 16 | 0.2388% | $60.00 | 5.26% | 16 | 0.5404% | $56.00 | 1.06% | ||
| 17 | 0.0890% | $70.00 | 17 | 0.0506% | $70.00 | 17 | 0.1306% | $65.75 | |||||
| 18 | 0.0144% | $80.00 | 18 | 0.0076% | $80.00 | 18 | 0.0223% | $75.50 | |||||
| 19 | 0.0015% | $90.00 | 19 | 0.0007% | $90.00 | 19 | 0.0024% | $85.25 | |||||
| 20 | 0.0001% | $100.00 | 20 | 0.00003% | $100.00 | 20 | 0.0001% | $95.00 |
Once again Lessons learned from the above charts are the longer one plays the less probability you have of winning. So keep your gambling sessions towards the short side and when you quit while you are ahead the better off you are on keeping your winnings.
Lesson Two is still to choose between bets that have lower house edges so you can have a better chance at winning
the most $$$ when you do win and losing the least $$$ when you do lose since your average loss will be less than bets that have a higher house edge.
The higher the house edge the higher the total
% of losing outcomes and the lower the total % of winning outcomes as the total amount of bets increases. The house edge can not and does not change the probability of any single outcome.
House Edge in the Short Run
Understanding Variance and Confidence Limits. For playing in the short run variance is more important than the house edge.
Where is the HOUSE EDGE? The pass line house edge is 1.41%
The expected value of 32 straight $5 pass line bets is a loss of $2.26.
Chart shows a different story. The most common outcome is to break even.
The 3 most common outcomes are 1. you break even, 2. you lose $10 or 3. you win $10.
In the short run the house edge or the expected loss may not always be the average outcome. Instead a player would want to know the winning and losing ranges or the confidence limits that come from the standard deviation of the bet.
The lines on the graph shows the upwards shift that comes from the pass line bet's actual winning probability.
Winning probability of 49.3% vs. a Losing Probability of 50.7%
The expected value of 32 straight $5 coin toss bets is a loss of $0.00.
Look what the chart shows. Exactly right. The most probable outcome is to break even.
The 3 most common outcomes are 1. you break even, 2 and 3 you win $10 or you lose $10.
The lines on the graph shows no shift in favor of either winning or losing probabilities since a coin toss is a 50-50 bet.