Vegas -
I played some Slots. Actually I put a dollar in the machine, bet a penny, and won $4.25. At this point I quit and collected by $5.25 thinking my return could never be higher.
The Gamblers Ruin problem goes back to the 1600s. If 2 gamblers bet back and forth a dollar you can calculate the probability that one will go bankrupt. I listened to Joe Blitzstein's lecture on this on his Harvard stats course at Itunes U
The important parameters are that gambler A has i amount of money and gambler B has N-i
The probability of A winning a round is p and of B winning is q which = 1 - p
If p = q, fair odds, then the odds of A winning = i/N or what fraction of the wealth A has.
The Probability of A winning = 1-(q/p)^i/1-(q/p)^N with unfair odds.
if p = .49 and i=N-i that is A and B have equal amounts of money.
then if N = 20 the chance of A winning is .4
N = 100 the chance of A winning is .12
N = 200 the chance of A winning is .02
In Las Vegas the casino has more money and games are more unfair
Since the probability of A winning + the probability of B winning is 1 there is no chance the game goes on forever.
In slots it is hard to figure out the odds of winning a hand as you can read below.
From Wikipedia
The return to player is not the only statistic that is of interest. The probabilities of every payout on the pay table is also critical. For example, consider a hypothetical slot machine with a dozen different values on the pay table. However, the probabilities of getting all the payouts are zero except the largest one. If the payout is 4,000 times the input amount, and it happens every 4,000 times on average, the return to player is exactly 100%, but the game would be dull to play. Also, most people would not win anything, and having entries on the paytable that have a return of zero would be deceptive. As these individual probabilities are closely guarded secrets, it is possible that the advertised machines with high return to player simply increase the probabilities of these jackpots. The casino could legally place machines of a similar style payout and advertise that some machines have 100% return to player. The added advantage is that these large jackpots increase the excitement of the other players.
The table of probabilities for a specific machine is called the Paytable and Reel Strips sheet, or PARS. The Wizard of Odds revealed the PARS for one commercial slot machine, an original International Gaming Technology Red White and Blue machine. This game, in its original form, is obsolete, so these specific probabilities do not apply. He only published the odds after a fan of his sent him some information provided on a slot machine that was posted on a machine in the Netherlands. The psychology of the machine design is quickly revealed. There are 13 possible payouts ranging from 1:1 to 2,400:1. The 1:1 payout comes every 8 plays. The 5:1 payout comes every 33 plays, whereas the 2:1 payout comes every 600 plays. Most players assume the likelihood increases proportionate to the payout. The one midsize payout that is designed to give the player a thrill is the 80:1 payout. It is programmed to occur an average of once every 219 plays. The 80:1 payout is high enough to create excitement, but not high enough that it makes it likely that the player will take his winnings and abandon the game. More than likely the player began the game with at least 80 times his bet (for instance there are 80 quarters in $20). In contrast the 150:1 payout occurs only on average of once every 6,241 plays. The highest payout of 2,400:1 occurs only on average of once every 643=262,144 plays since the machine has 64 virtual stops. The player who continues to feed the machine is likely to have several midsize payouts, but unlikely to have a large payout. He quits after he is bored or has exhausted his bankroll.[21]
