Is Probability really consistent with our world?

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?

• The probability for the sixth toss is indeed 0.5 if you look at it in an isolated way. However, if you look at the result of a series of six tosses (counting the number of heads and tails) then the probability for the event "six heads" is different to those for "five heads, one tail". – Dr_Be Feb 10 '16 at 7:38
• $0.5$ probability of heads. You said they were unbiased. That is the definition of unbiased. – Zubin Mukerjee Feb 10 '16 at 7:38
• Lets say we do a little tiresome experiment, We toss 5 coins and when we get 5 heads, only then, we do another coin toss and we write down the outcome of the sixth toss, and lets say we do this 1000 times. What will the result be then? How many times we get tail on the sixth toss? – mobifz96 Feb 10 '16 at 7:42

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.

• Very good explanation. Now things are clearer than ever. – mobifz96 Feb 14 '16 at 9:29

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.

• Yes its 0.5, I took a lot of time to swallow that one. And probabilistic approach to express our world is actually really a bad idea unless we have trillions of things. – mobifz96 Feb 13 '16 at 5:09

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.

• Legit but when we see six tosses as a system. last one should be very high probability of tail, because when we do the experiment thousand times, the overall probability should be 0.5, hence the sixth toss should lean to the tail side. Ain't that the truth? – mobifz96 Feb 10 '16 at 7:49
• When you toss 1000 times, You need to get all first 5 heads first. Then you need to count no. of tails. Toss 10,000 times and take the ... let us say 100 times when you get 5 heads in the first 5 turns. Then count the number of tails on the 6th coin. It will be 50. – Win Vineeth Feb 10 '16 at 7:51
• @mobifz96 I mentioned it in my previous comment above. The probability for the sixth toss is 0.5 regardless of what happened before (assuming of course independent tosses). It's a totally different story if you look at the result of all(!) six tosses. Here six times head has a lower probability (1.5%, $1/2^6$) than five times head and once tail. This becomes clear by looking at the number of different ways leading to the respective outcomes. – Dr_Be Feb 10 '16 at 7:58
• Did a lot of further reading and now the answer is obvious its 0.5 and that probability is really inconsistent with our world. – mobifz96 Feb 13 '16 at 5:08