Casino games are built around probability. To demonstrate this, let’s look at American roulette.

Roulette has 38 spaces (18 Red spaces, 18 Black spaces, 1 Zero Green and 1 Double 00 Green). Instead of betting on a specific number to be hit, we’ll place a bet on red, meaning we’ll be paid out if the pill (the ball) lands on any red number.

\[P(win) = {18 \over 38}\]

\[P(lose) = {20 \over 38}\]

What is the casino expected to win on a bet like this?

Using our previous example on Roulette, let’s look at what the Casino would take if our player bet $1.

\[Expected(X) = {18 \over 38}(1) + {20 \over 38}(-1) = -0.0526\] As we see above, we multiply the “win” ratio with 1 dollar, and the “lose” ratio with -1 dollar, giving us -0.0526. In this case, negative “Expected” means positive gain for the casino. For this example, every 1 dollar of this type of bet has an expected value of 5.26 cents, which we can call “House Edge”(A house edge of 5.26%).

Simply, House Edge is the expected percentage a casino will win from a player’s wagers over time. Some games have extremely low house edge, like Blackjack at 0.5%.

We alluded to earlier that for every $1 bet, the casino was expected to win 5 cents. This calculation is called “EGR” or estimated gaming revenue.

\[EGR = Wagered * House Edge\] Lets scale our previous example. If we were to bet $75, but only on 14 (because it’s your lucky number).

\(P(win) = {1 \over 38}\) \(P(lose) = {37 \over 38}\)

Lets use our new EGR Formula. How does this compare to our Expected Formula?

\[EGR = 75 * 0.0526 = ~ 3.945\]

Lets break down the spin. Remember, the payout of 1 singular number isn’t double like our previous example (bet a dollar to win a dollar), it’s 35 times the bet for a singular number.

\[Expected(X) = {1 \over 38}(2625) + {37 \over 38}(-75) = -3.947\] As expected, the math checks out! Why is it only ~4 dollars and not the full $75 lost? This formula is the expectation over many bets, and not just the one spin. Actual casino results can vary significantly from expected results in the short term. As the number of wagers increases, however, actual results should begin to more closely resemble expected results.

Let’s set up a a mini roulette simulation and see how it compares to the math. For this example, we are going to continuously pick 1 number (our lucky number 14), 10 times using the probability of 1/38. We’ll also continue our bet of 75 dollars.

spin_count <- 0
lucky_number <- 14
bet <- 75
winnings <- 0

while (spin_count < 10){
  number_picked <- sample(1:38, size = 1)
  
  if(number_picked == lucky_number){
    winnings <- winnings + (bet*35)
  }else{
    winnings <- winnings - bet
  }
   spin_count <- spin_count + 1
}
winnings

## [1] -750

On the previous slide, you most likely saw -$750. But it could have easily been 1,950 dollars for the player. Why was it so far off? Well lets run it again but this time 10,000 spins.

Variance causes actual GGR to fluctuate around expected GGR. As the number of wagers increases, the casino’s actual return relative to the total amount wagered should tend toward the theoretical house edge. With our findings, lets compare Actual House Edge % vs. 5.26%.

As the number of spins increases, the observed house edge begins to stabilize toward the theoretical 5.26% house edge.

Even with the exact same game, bet size, and number of spins, actual casino revenue can vary significantly. Let’s simulate 1,000 separate sessions of 1,000 roulette spins.

Although expected GGR is about $3,945(the dotted black line), individual simulations can result in very different outcomes due to variance.Some players looked to be down 40k while others were up 40,000.

Casinos use probability to determine the expected value of their games. However, actual GGR (Gross Gaming Revenue) can vary significantly from EGR in the short term due to variance. As the number of wagers increases, actual casino hold tends to stabilize toward the theoretical house edge.