Math Is Fun Forum

gnitsuk

Member

Registered: 2006-02-09

Posts: 121

Probability

Hi,

Could I ask, would folks agree with my calculation below:

It is to do with tossing N coins.

By an "experiment" I mean the act of tossing a N coins.

So:

n = Number of experiments performed.

P = Probability of getting N heads in a row after n experiments.

I wish to derive a formula that gives n as a function of P and N.

Here's the reasoning:

On the first experiment, the probability of getting N heads in a row is:

So the probability of not getting N heads in a row is:

If one actually gets N heads in a row at this point then one stops. However, if one does not yet have N heads in a row then one continues.

I'm visualizing this as a probability tree diagram where at each splitting of the tree there are two paths, one with probability

and the other with probability

So with this setup, one could ask, what is the probability of obtaining the N heads in a row in two experiments. Well, for this to have happened either "the first experiment would have succeeded" OR "the first experiment would have failed and the second experiment would have succeeded" in producing N heads in a row. So the probability would be:

In this way, we can form an infinite sum which gives us the probability of obtaining the N heads in a row after n experiments, it is:

This is geometric progression and so we can find the sum of the first n terms as:

which simplifies to:

which gives:

So is this correct?

Well we can ask what it gives for the various possibilities below:

1) How many experiments should I perform such that the chances of me getting all heads with 10 coins (N = 10) is 0:

Formula gives

That makes sense.

2) How many experiments should I perform such that the chances of me getting all heads with 10 coins (N = 10) is 1 in 1024:

Formula gives

That makes sense.

3) How many experiments should I perform such that the chances of me getting all heads with 10 coins (N = 10) is 1:

Formula gives

That makes sense.

4) How many experiments should I perform such that the chances of me getting all heads with 10 coins (N = 10) is 0.9999:

Formula gives

(rounded up to nearest whole number of experiments)

That makes sense?

My question is, would folks agree with the correctness of this formula?

Thanks,
Mitch.

Last edited by gnitsuk (2014-12-15 23:07:47)