Say we have 6 unbiased coins, We toss 5 coins and get 5 heads. Then what is the probable outcome of the sixth toss? Mathematically every new and discrete event should be independent of the results of the previous independent events, So the probability should be 0.5 for either side. Question is that Which one is true, A high probability of a tail or 0.5?
Let's toss a series of six unbiased coins. At the start of this experiment, there are $2^6 = 64$ possible outcomes. Two of these outcomes are $HHHHHH$ and $HHHHHT$. Those two outcomes are equally likely (or unlikely) to occur.
Before any coins are thrown, there is only a $1/32$ chance that we will reach a point in time when the first five tosses are all heads. If the first coin lands heads, that probability goes up to $1/16$. But if the first coin lands tails, that probability goes to zero.
If we happen to get five heads in the first five tosses, as unlikely as that event is, we will have eliminated almost all the possible sequences of six tosses that we had at the beginning. For example, there is now zero probability of $HHHHTH$ or $HTTHHT$.
The only two possible results remaining are $HHHHHH$ and $HHHHHT$. Back when we started the experiment, those two outcomes were equally likely to occur. What would make one of them now more likely than the other?
If the coins are unbiased then the two outcomes are still equally likely, but their total probability now is $1$ and their individual probabilities therefore are $1/2$.
The one reason not to assign equal probability to head and tails after five heads is if there is doubt about the premise that the coins are truly unbiased. The obvious doubt is that you may suspect that someone has slipped a few two-headed coins into the coins you have to flip. But if that has happened, it makes the next toss more likely to be heads, not tails.
Which one is true, a high probability or $0.5$?
If indeed every new and discrete event is independent of the results of the previous independent events, then the probability is indeed $0.5$.
So the only question remaining is - are those events really "physically" independent of each other?
I don't think that science has a decisive answer for this question at present.
You might want to read a little bit about the Butterfly Effect and the sensitive dependence on initial conditions in which a small change in one state can result in large differences in a later state.
It's surely 0.5. By no chance is the probability of tail going to increase. We tend to look at it that way only because, we have a feeling that successes and failures don't come continuously, but come pair-wise. That is a notion of probability where all the partition events are exhausted before any event starts repeating. However, actually, it doesn't happen so. The answer to your comment is once again the same . 500 tails.