Will you ultimately win if you keep on gambling?

Gambling gurus and math mavens all say that in games of independent trials, including nearly everything in the casino except blackjack and its derivatives, and even there to a great extent, probabilities don’t vary between rounds. If your chance of a result is one out of five the first time you try, it’s one out of five on your second, third, fourth, 10th or any other shot as well.

This contrasts with what solid citizens imagine intuitively. Namely, that the more often they play, the more likely any given result is to eventually occur.

Who’s right, the pundits or the players? Both – depending on what each of them means.

As an illustration, picture a hypothetical gamble involving four equally-possible outcomes. On each try, the chance of winning a bet on any of them is one out of four (25 percent) and of losing is the other three out of four (75 percent). But, inquiring minds might want to know the probability of winning once, then quitting, if they’re willing to keep going for up to those two, three, four, and 10 attempts? The answer is 43.750 percent at the second, 57.813 percent at the third, 68.359 percent at the fourth, and 94.369 percent at the 10th. Notice though, that although prospects increase with the number of tries a player is willing to make, the rate at which chance improves is a diminishing quantity. Going to two rounds, it doesn’t double, from 25 to 50 percent, but rises by 18.75 percent, from 25 to 43.75 percent. Moving from two to three rounds, the increment is 14.06 percent. From three to four tries, the gain is 10.55 percent.
Two patterns emerge from these figures. 1) The more you play, the greater your chance of eventually winning. 2) The increase in the probability you’ll win diminishes the further you go. And, the way the gain in your chances keeps declining, you may get ever closer to 100 percent probability of hitting, but never quite reach it. These factors are of particular interest to casino aficionados who pursuit big paydays on the machines. Jackpots are elusive. But do frequent players have a better chance than occasional recreational gamblers?

Pretend your game is video poker, at which the jackpot probability is one out of 40,000, or 0.0025 percent. Every time you push the button, your chance of fulfilling your fantasy is the same 0.0025 percent. But what’s your chance of eventually winning if you play a lot?

Say you’re planning to gamble for up to four hours, quitting when the bus pulls out or you hit a jackpot. Assume you average a spin every 10 seconds, 360 per hour. Your chance of missing on the first 359 and hitting on the 360th, at the end of the first hour, is way up to 0.8608 percent – one out of 116. That of agony on the first 719 followed by ecstasy on the 720th, at the end of the second hour, leaps to 1.647 percent, one out of about 61. Success on the 1,080th after 1,079 misses, jumps in at 2.366 percent, one out of 42. And the probability of missing for four solid hours and scoring on your last spin – number 1,440, escalates to 3.024 percent, one out of 33.

The difference between the two notions of chance after repeated trials can be reconciled in terms of conditional and joint probability. In the present context, conditional probability is the chance that, given the previous occurrence of one event, another will happen. Here, this might be the likelihood that after four straight misses, the next try will hit. Joint probability is the chance that multiple events will occur together. An example might be the chance of four misses then a hit.

In games of independent trials, the probability of a specified result on any round is conditional. The chances of what already transpired are of no interest. The uncertainty is gone and they’re matters of history, not probability. Picture a fresh 52-card deck. If you draw one card, the probability it’s an ace is four out of 52 or 7.692 percent. Say you draw an ace, replace it, then draw again. The chance of an ace on the second round is still 7.692 percent.

When trials are not independent, the probability of a specified result on any round is joint. The chances of what happened previously are relevant. Say, as above, you draw a card from a fresh deck. The probability of an ace is 7.692 percent. If you don’t put back the first card before drawing the second, two situations may pertain. With anything other than an ace on the first try, the chance of success on the second is four out of 51 or 7.843 percent. When the first card is an ace, the chance of another as the second is three out of 51 or 5.882 percent.

Of course, perhaps you think rounds at your favorite game aren’t independent. Rather, that biases or patterns occur that you can exploit in deciding on what or how much to bet. If so, you may have been reading too many books or watching too many movies from the ‘30s or ‘40s. The immortal inkster, Sumner A Ingmark, put it like this:
Complaints of a conspiracy
Oft border on a heresy.