The odds of seeing a series of n outcomes

[deisik]

Let's toss some coins here

And start with tossing a coin for once only. We know that our chances are 50/50, i.e. we get either heads or tails in a series of just one toss. As simple as it gets. Let's now consider a series of two tosses, either all heads or all tails. We also know that the odds of such a series are 0.5x0.5=0.25. Seems as simple, right?

What it means is that if we toss a coin a billion times, we will see around 250 million series of at least two heads or two tails. So far so good. Let's then increase the length of a series to, say, 10 heads or tails. Now, the odds are 0.5¹⁰, which is 0.0009765625, and this is where things start to get complicated

Again, if we make a billion tosses, we will see a streak that long (or longer) every 1/0.0009765625=1024 tosses on average. And here comes the interesting part. What are our chances to see this series in the first 1024 tosses? It should be evident that the chances are not 100%, i.e. we are not set to encounter such a streak in the first 1024 tosses at all

But what are the odds then? I think it is impossible to tell exactly due to the random nature of tosses. So we could simply say that the odds are 50% but only to show that there is no certainty, while any other percentage would effectively tell us something about the distribution, thereby violating our assumption about the randomness of tosses

What's your take on this?

[deisik]

Okay, I've been thinking about that, and here're my thoughts

So we could simply say that the odds are 50% but only to show that there is no certainty, while any other percentage would effectively tell us something about the distribution

If we assume that the probability of seeing a streak of n heads or tails is equal to 50% within N tosses (as we are talking about coin flipping here), we should then accept that the odds of seeing it in the next N tosses are the same because tosses are independent, and we can't and mustn't distinguish between the first sequence and any other sequence of N tosses

However, that would mean that the probability of seeing that streak in these two sequences (i.e. N*2 tosses) is 100%, which is obviously a wrong assumption. It might well be over 90% but still not certainty. Then we necessarily come to the conclusion that the probability of seeing the streak in the first sequence of N tosses is in fact below 50%

On the other hand, it could be argued that we can't really assess these probabilities as such assessment cannot be made in the same way we can't assess the probability of an event that has already transpired. In the latter case, we can't even say that its probability is 100% since there's no probability

This is important to keep in mind

[Timelord2067]

You'd love statistics.  Calculating the probability of one side over the other - there's a tale of a POW during WW II who tossed a coin over ten thousand times and concluded that if the coin is tossed a great number of times the chances of heads or tails will work out to be 50%/50%.  There may be streaks where one side or the other is favoured for a short while, however over time that error will correct itself.

[deisik]

You'd love statistics.  Calculating the probability of one side over the other - there's a tale of a POW during WW II who tossed a coin over ten thousand times and concluded that if the coin is tossed a great number of times the chances of heads or tails will work out to be 50%/50%.  There may be streaks where one side or the other is favoured for a short while, however over time that error will correct itself

Well, this topic looks into a different aspect of coin tossing

It is not about the probability of either outcome, heads or tails, as it is known for certain to be 50% on average. Put differently, it is either heads or tails, while the odds themselves are equal, and no third option is possible. So it is necessarily 50%. However, I'm trying to assess a different type of probability here, i.e. the odds of actually seeing, say, heads in the next 2 tosses (as 1/0.5=2) or any other particular streak of tosses, for that matter (say, 2 heads in 1/0.25=4 tosses). As it turns out, the odds are less than 50%

[Timelord2067]

As it turns out, the odds are less than 50%

Correct - the chances diminsh as the number of consecutive tosses occurrs.

First toss ½ (being one of two sides) i.e. 50%

Second toss ½ X ½ = ¼ i.e. 25%

Third toss ½ X ½ X ½ = ⅛ i.e. 12.5%

The probability diminishes very quickly beyond that.  Results are multiplied, not added together.

[deisik]

As it turns out, the odds are less than 50%

Correct - the chances diminsh as the number of consecutive tosses occurrs

Let's take a simplest example of two tosses

The probability of 2 heads (HH) in a row is indeed 0.5x0.5=0.25 in 2 tosses. The other possible outcomes are HT, TH, and TT. And if we make 1 billion tosses, we will come pretty close to that probability (0.25 for every couple of tosses). So this point is not challenged here. However, it is not at all set in stone that after we make 2x4=8 tosses (4 possible outcomes in 2 tosses), we are going to see the HH streak for certain even once. But if it is not a certainty, what are the odds then given that they should be less than 50% anyway?

[Ucy]

Trying to understand what the "series" means in coin tossing world. The "series of one toss" make the question abit difficult to understand. Is series the "rolling in the air" a coin goes through when tossed?
Hopefully, I'll get an answer and reply today or tomorrow.

[deisik]

Trying to understand what the "series" means in coin tossing world

It is the same as everywhere

A series is just a certain number of consecutive, i.e. following one after another, tosses (aka outcomes). In dice, that would be rolls. "A series" of one toss is just that one toss, i.e. either heads (H) or tails (T). Then, a series of two tosses can be HH, TT, HT, and TH. A streak generally refers to a series of same outcomes like HHH in coin flipping or LLL in dice

I think we can further reduce the task to just one toss. The chances of seeing either heads or tails are 0.5 in one toss, and they would be that on average after, say, 1 million tosses. Statistically, it means that in every two tosses we are going to see exactly one head and one tail (as the odds of either are 0.5)

With that said, though, there is no certainty that we are actually going to see either in any given series of 2 tosses. The idea is that in practice there will be long streaks, and thus we can't say that we should always see a head in the next two tosses, or any two tosses, even if the probability of heads is 0.5. But if it is not 100%, what are the chances then?

[Timelord2067]

You really need to brush up on statistics and then look to probability.  I've already answered your question, but if you want to toss a Billion times (short Billion, or long Billion - it matters not) then go right ahead.

Try Roulette, the Red and Black pay 1:1 and the outcome (asterisks) is about 50%/50% - note the asterisks - the green zero (or double zero) is the house edge.  Go to a live Casino and watch how many times one colour or the other comes out in a row.  Not more that four or five times in a row.