Confusion about probabilities.

Question

Ok so far what I understand is this let's say... Having to draw a card from $52$ card-deck its probability is of course $\dfrac1{52}$. Now the probability to say that I will keep drawing this same card $10$ times of course it is $\dfrac1{(52)^{10}}$ because now I have a lot more possibilities. Now my question is this. I also know that drawing a card at a specific time has the same probability of $\dfrac1{52}$ no matter what you drew before. Is it wrong to assume that the fact that you drew $9$ times the same card it is most likely that it won't happen again. Because as a whole fact the probability is $\dfrac1{52^{10}}$. Let's consider that you don't know what you drew before. It is somehow very counter intuive. I know the math I think but I can't dygest it. I mean when we play games that have the "random" element in them I mostly do well because I go with my intuition like they say "lightning doesnt hit the same spot twice".

Answer 1

The question it seems you are asking about can be phrased as follows:

Given that I have selected the ace of spades 9 times in a row (with replacement), what is the probability that, upon drawing another card at random, I will draw the ace of spades again?

The traditional mathematical answer is that, assuming that all cards have the same basic properties, the chances of drawing any two cards are the same. It would follow that the probability of drawing the ace of spades is necessarily 1/52, regardless of what had occurred previously. The deck does not "remember" what has happened before, so the probability of drawing a particular card, which is a property of the deck, should not change.

It is a common mistaken belief that if one outcome occurs with unusual frequency, the odds must be shifted in favor of other outcomes in order for the frequencies to "balance out". This mistaken belief is referred to as the gambler's fallacy.

That being said: one might say that certain outcomes are "suspicious" and perhaps reveal that the system isn't "fair" as you may have assumed. For example, if you rolled a six-sided die and got a 1 ten times in a row, you might think that perhaps the die is loaded.

It is often useful to adjust your guess of what the underlying probabilities are with a Bayesian update.

Answer 2

You're right, given that you have drawn $9$ times the same card, it is unlikely to draw the same card again, just as it is if you had drawn that card only once or one hundred times, namely $\frac{1}{52}$. I think where your intuition goes wrong is that you fix your attention on one specific card if it appears a lot. Given that one specific card (or lightning spot) the chance is pretty low to get it if you compare it with its complement: not that card. However, if you compare it with the probability to getting any other card, they are of course all the same.

Answer 3

In fact, if I draw the same card from a deck 9 times, I would think that there is some kind of trick, and that next time I would draw the same card again.

But if I'm sure that the deck is like expected, that is, with 52 different cards that have all the same probability of being drawn, then there is no doubt: no matter what have you drawn before, the probability is always the same. The skepticism that some people shows before this fact is perhaps because of the very little probability of that a random experiment yields many times the same result.