Can the probability of the same random events be predicted when the probability of each event is known?

Question

This question relates to "Gambler's Fallacy". For the sake of this question, please assume that by using basic strategy that the probability of winning any single hand of Blackjack is almost 50/50. In this scenario, I know that the outcome of any single hand is unrelated to the outcome of the series of hands that preceded it. However, can a single desired outcome be predicted over a series of future hands? If I am not mistaken, the odds of ALL the player's hands being losses over a future series looks something like this:

(1) Hand: 50/50 (0.5)
(2) Hands: 75/25 (0.25)
(3) Hands: 87/13 (0.125)
(4) Hands: 94/06 (0.0625)
(5) Hands: 97/03 (0.03125)
Etc....

If this is true, might one conclude that the probability of winning at least ONE hand in the next four consecutive hands, for example, is about 93%?

If this is also fallacious thinking, please explain.

Answer 1

If you're accepting the thought that every hand is random and that the probability of winning any hand is 1/2, then you're correct.

This would be a binomial distribution. The probability of winning at least one hand = 1 - (the probability of losing all four).
1 - (1/2)^4 = 1 - 1/16 = 15/16.

The only fallacy to the actual process is that the probability of winning subsequent hands after the first isn't going to be exactly 1/2, as the cards that were used are gone from the deck and change the odds for the rest of the deck. But if you're looking for a good ballpark figure, you're on the right track.

Answer 2

No..... its a contradiction. "Random" MEANS just that, and if you have "the same" random events, then its not random. This can be applied to most state Lotto' as well.....

Using a set of SIX numbers out of 52-53 etc. there is ONLY a certain number of combinations possible!!! What happens is that the computers running those Lotto' produce over and over and over duplicate wrong combinations, hence the odds being 6 million or whatever to 1 of winning.

Cards games such as Blackjack, Poker etc. operated as a business (ie: Casinos etc.) are designed the same way, to repeat the wrong sequence of cards over and over.

Otherwise, you're right.... playing such games in an environment where its NOT designed to win the "house" money, there's only a set number of possiblities within the deck of 52 cards.

Answer 3

No, its not fallacious. But it does not help you predict anything really.

Each hand played still has a 50% chance of winning. But the chances of playing a large number of hands randomly without a win is obviously going to be low - though you are now talking about theprobability of a combination of results not a single one.

So, imagine you have played 10 hands without a win. What is the chance the next hand will be a win. Well, its still 50/50.