How Many Times Need to Flip Coin to Get 100x Head (or tail) in Row?

[mathhead!]

50% chance to get head or tail with one flip.

So, let's guess how many times need to flip the coin until we can get Head 100x in row, until someone will share with us how to calculate this.
My guess is 15 million times. What do you think?

 

[Dr.Peterson]

Are you looking for certainty, or for an expected number of flips, or something else?

You need to define what you are asking carefully, which requires a certain level of understanding of probability.

 

[Steven G]

Here is how you find the probability of flipping 100 heads in a row.

You can flip a coin 100 times and get 100 heads in a row. What is the probability of this happening?
You can flip a coin 101 times and the last 100 flips is the 1st time you had 100 heads in a row. What is the probability of this happening?
You can flip a coin 102 times and the last 100 flips is the 1st time you had 100 heads in a row. What is the probability of this happening?
You can flip a coin 103 times and the last 100 flips is the 1st time you had 100 heads in a row. What is the probability of this happening?
.....
What do you do with these numbers?

 

[mathhead!]

Are you looking for certainty, or for an expected number of flips, or something else?

You need to define what you are asking carefully, which requires a certain level of understanding of probability.

What do you mean by certainty vs expected number of flips? I am looking for probability of this scenario.

For example, dice has six angles, to get number 1 we have to throw dice six times on average. So how many times (probability) we have thrown dice to get same number two times in row? That I am looking for.

 

[mathhead!]

Here is how you find the probability of flipping 100 heads in a row.

You can flip a coin 100 times and get 100 heads in a row. What is the probability of this happening?
You can flip a coin 101 times and the last 100 flips is the 1st time you had 100 heads in a row. What is the probability of this happening?
You can flip a coin 102 times and the last 100 flips is the 1st time you had 100 heads in a row. What is the probability of this happening?
You can flip a coin 103 times and the last 100 flips is the 1st time you had 100 heads in a row. What is the probability of this happening?
.....
What do you do with these numbers?

I don't know what is the probability, that's why I posted this topic

 

[Dr.Peterson]

What do you mean by certainty vs expected number of flips? I am looking for probability of this scenario.

For example, dice has six angles, to get number 1 we have to throw dice six times on average. So how many times (probability) we have thrown dice to get same number two times in row? That I am looking for.

The trouble is that "how many times" is not a probability! It's a count.

My question is, do you want to know how many times you would have to throw to be SURE of getting a certain number? You can never be sure of an event in probability. So you might instead mean that you want some set probability that it will happen, say 95% certainty. In that case, you have to specify that number to get an answer.

Or, you might want to know the expected value -- that is, "on average", how many times would it take? That's a different question, and has a specific definition in probability theory. Since you used that term, "on average", here, that is likely what you want. That is not what you asked at first.

Anyway, it takes some work to derive the answer. Here is one page I found about it: https://courses.cit.cornell.edu/info2950_2012sp/mh.pdf

 

[mathhead!]

The trouble is that "how many times" is not a probability! It's a count.

My question is, do you want to know how many times you would have to throw to be SURE of getting a certain number? You can never be sure of an event in probability. So you might instead mean that you want some set probability that it will happen, say 95% certainty. In that case, you have to specify that number to get an answer.

Or, you might want to know the expected value -- that is, "on average", how many times would it take? That's a different question, and has a specific definition in probability theory. Since you used that term, "on average", here, that is likely what you want. That is not what you asked at first.

Anyway, it takes some work to derive the answer. Here is one page I found about it: https://courses.cit.cornell.edu/info2950_2012sp/mh.pdf

Probability to get 10 heads in row needs 2046 coin flips. So how many coin flips need to get 100 heads in row and how to calculate that?
Here is some article about it: https://nrich.maths.org/6954/solution#:~:text=Jungsun: There is an 1,be multiplied for 10 times.

 

[Dr.Peterson]

Probability to get 10 heads in row needs 2046 coin flips. So how many coin flips need to get 100 heads in row and how to calculate that?
Here is some article about it: https://nrich.maths.org/6954/solution#:~:text=Jungsun: There is an 1,be multiplied for 10 times.

I take it this is the context of your question? It would have been helpful to quote the claim, so we'd get the wording right sooner: "it takes, on average, 2046 flips to achieve 10 heads in a row". (The wording is not "Probability to get 10 heads in row", which is more or less meaningless.)

Did you look at the link I provided? On page 2, it gives a formula: [MATH]2^{n+1}-2[/MATH].

For n=10, this gives [MATH]2^{11}-2 = 2046[/MATH], just as your source claimed.

For n=100, what do you get? (It's a lot more than 15 million.)

 

[Steven G]

I don't know what is the probability, that's why I posted this topic

We do not give out answers on this forum. We give out hints to arrive at your answer. I told you what probabilities you need to compute. OK, so you do not know how to do that.

Let us consider the 1st probability. What is the probability of flipping a tail followed by 100 heads? The flipping of a coin is independent of one another. So you just multiply probabilities. What is the probability of flipping a tail 1st? Then a head? Then another head? ... Then another head?

 

[mathhead!]

Okay, I just learned I can calculate for example 3 heads in row like this: 0.50*0.50*0.50=0.125. That is 12.5% possibility to get 3 heads in row. 100 / 12.5=8.
I need approximately 8 flips in average to get 3 heads in row. Is this calculation right?

 

[Steven G]

Okay, I just learned I can calculate for example 3 heads in row like this: 0.50*0.50*0.50=0.125. That is 12.5% possibility to get 3 heads in row. 100 / 12.5=8.
I need approximately 8 flips in average to get 3 heads in row. Is this calculation right?

I am not sure why you say that 100/12.5 = 8 and then say it is approximately 8.

I do not think what you are saying is correct.

If you flipped 3 coins and observed if you got 3 heads in a row and then flipped 3 more coins and checked if you got 3 heads in a row etc, then yes it will take on averages 8 tosses of three coins to get 3 heads in a row.

However you question is quite different. For example if you flipped thhhth then you should say that you tossed 3 heads in a row.

However if think of thhhth as thh and then hth you did not toss 3 heads in a row. This thinking here is what you did in your last post. But it is the wrong way of thinking about it. You should consider thhhth as having flipped 3 heads in a row.

Please post back and let us know if you are seeing the difference I stated above.