Suppose we have a fair coin. We are going to define an experiment.
A single coin is tossed infinite times.
Now, imagine it this way - if a coin is tossed $n$ times, the probability of getting $\frac{n}{2}$ heads decreases as $n$ increases. However, what happens when $n$ tends to $\infty$ ? Many sources tell me, if a coin is tossed infinite times, I would get equal heads and tails. However, when it is tossed a finite number of times, I see that the probability of getting a equal number of heads and tales decreases, as the number of tosses increase. How does it suddenly increase and become $50$ percent, when the number of tosses become infinite ?
For two tosses, probability of getting equal number of heads and tails is $0.5$. For 4 tosses, this becomes $0.375$. For a hundred tosses, it becomes $0.08$. We can clearly see, even though equal head and tail is the most probable, it is less likely to occur.
However, when we take infinite tosses, we are expected to get equal heads and tails, so the probability jumps up to $0.5$ again ?
How is this possible ? Is there something wrong with my reasoning ?
Having an equal number of heads and tails is not an event that we expect. Actually there is another branch of mathematics known as design interference. If you would get more frequently exactly the same number of heads and tails, that would violate the basic assumptions i.e. the axioms of probability that the things are happening independently.
When we say we expect that the number of heads and tails is equal, we actually have in mind a normal distribution where after as you call it an infinite number of attempts you would create a bell like shape that is centralized around equal number of heads and tails and with a very precise shape of the percentage of other possible outcomes. So precise, actually, that any deviation inevitably means a rigged coin or loaded die or whatever other crooked mechanism has been employed or was simply present.
Without any doubt.
Notice that when you talk about an infinite number of attempts, you cannot talk about an equal number of heads of tails and compare it with other outcomes. Can you say that the outcome was one head less towards infinity? Yet, that is an event as well that you could track. What would you find? That the number of such precise outcomes has a probability equal to $0$.
For this reason, when we talk about infinity we use other means for example a histogram that would say a percentage of events or something similar that will not vanish as we go towards the infinity. For example, the percentage of outcomes where the total number of tails is within $20$% of the number of heads. Then and only then you can talk about the most probable set of events, but not one single event.
Because design interference is similar to probability and share many of its principles, yet it still reveals a gap in our understanding of probability, you have to be really careful when you assume that things are happening totally by chance. Otherwise you could easily confuse things and forget to check if your test is really random or it was influenced by something, like a rigged die.
This is the reason why you are going to be kicked out of a casino as soon as the pattern of wins even resembles something that is not pure luck even when it is pure luck. $2^{36}$ is a lot of money even if expressed in cents, and trust me I myself can trick any roulette to get at least $0.1$% of that if there would be no guards to kick me out forever as soon as they would notice the tricks I would use(, and 700.000 dollars is still a lot of money for one night).