Reliability of streak probabilities

Say i am tossing a fair coin 100 times.
The statistics say that I have a 25% chance of observing a streak of 3 or more of either heads or tails at the beginning of each toss.
How much can I count on this probability?
Say a streak of 3 or more only happened 6.25% of the time in my trial, can I calculate how many SDs from mean this outcome is?
Or can I count on seeing a streak of 3 or more being extremely close to 25% over 100 trials?

Quote: Bmayo319

Say i am tossing a fair coin 100 times.
The statistics say that I have a 25% chance of observing a streak of 3 or more of either heads or tails.
How much can I count on this probability?
Say a streak of 3 or more only happened 6.25% of the time in my trial, can I calculate how many SDs from mean this outcome is?
Or can I count on seeing a streak of 3 or more being extremely close to 25% over 100 trials?

AFAIK, the only "probability" to be relied upon is that a truly fair coin will, over time, land tails half the time and heads the other half.
There is no way to reliably predict when a "streak" will begin or how long it may last.
(Of course, I could be mistaken, but I'll stick with this until proven otherwise.)

If you're tossing a fair coin 100 times, you're going to have a streak of 3 or more during those 100 tosses.

I don't know if that's what you're asking though.

Quote: Bmayo319

Say i am tossing a fair coin 100 times.
The statistics say that I have a 25% chance of observing a streak of 3 or more of either heads or tails.

how did you arrive at that probability?
I get for heads or tails = 0.999999999095732 for at least one such run of either
1 in 1,105,866,845 of NOT seeing such a streak. It could happen.

for 10 flips I get: 0.859375 as the probability
not a certainty
for 3 flips I see 25%
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25% = 2/8 *100

Quote: Bmayo319

How much can I count on this probability?

if the probability of the event is known and is correct
you can count on it the same as the probability of a head or tail on the next flip is 50%

Quote: Bmayo319

Say a streak of 3 or more only happened 6.25% of the time in my trial,

now you are talking about X number of successes over N number of trials

flip a fair coin 10 times and see only 3 heads.
Is the probability of flipping a head now 30% or 50% as you may think it is?
this is the difference between a statistic and a probability
Probability is How likely something is to happen
or
Probability of an event happening = Number of ways it can happen / Total number of outcomes
http://www.mathsisfun.com/data/probability.html

But for a run of 3 by either may have happened 0 times.
that would be 0% and not 25% of the time

But your event is a run and that is a dependent event and not an independent event.
(A run of length 3 can never happen on just 2 flips)
so the math is different for the expected number of successes or the distribution of successes

as in 10 flips we can only see at max 3 such runs if we count that way to restart after 3 in a row
so now even the counting method used comes into play. and 3/10 *100 = 30%
and that is the max in 10 flips counting all the flips and not all the possible spaces where a run can happen.
now how we use the numbers becomes important.

you appear to be interested in Baccarat shoe probabilities and statistics
as an example
what is the probability that in one 8 deck Bac shoe Player will NOT see a run of 3 or more?

Imagine how much money you would make by knowing this was going to happen
or betting that it would happen when it looks like it could happen.

By the probability we know it can happen as that probability is >0
about 0.003709204 or 1 in 267 shoes on average for 0 runs of length 3 or more

but the number of times it happens over X number of shoes is still a random variable.

the percentage of times it happens will converge to the probability of the event, given a large number of trials or as the number of trials increases,
but not the actual number of times it happens does not converge to the expected value over X trials.
This is the Law of Large Numbers

Sally

Quote: Bmayo319

Say i am tossing a fair coin 100 times.
The statistics say that I have a 25% chance of observing a streak of 3 or more of either heads or tails.
How much can I count on this probability?
Say a streak of 3 or more only happened 6.25% of the time in my trial, can I calculate how many SDs from mean this outcome is?
Or can I count on seeing a streak of 3 or more being extremely close to 25% over 100 trials?

For any 3 flips, there is a 25% probability that they will all be the same.

So, be sure to count 4 in a row as 2 streaks (the streaks are (1, 2, 3) and (2, 3, 4)). In this case, over 100 flips, you have 98 possibilities for a streak, and so you should expect about 24 or 25 on average.

Variance is a little harder to calculate. Due to the overlapping nature of the streaks, they are not independent (following a streak, you have a 50% chance of another streak, but, following a non-streak, you have only a 1/6 chance of a streak). I don't see a quick way to calculate it -- I'm not even sure what the distribution would look like.

Quote: AxiomOfChoice

For any 3 flips, there is a 25% probability that they will all be the same.

So, be sure to count 4 in a row as 2 streaks (the streaks are (1, 2, 3) and (2, 3, 4)). In this case, over 100 flips, you have 98 possibilities for a streak, and so you should expect about 24 or 25 on average.

Your math is way off. MustangSally knows how to do this calculation. Out of 100 tosses there is actually a 33.791% probability of observing a streak of 7 or more of either heads or tails. The odds of observing a streak of 3 or more of either heads or tails is nearly 100%.

Quote: FinsRule

If you're tossing a fair coin 100 times, you're going to have a streak of 3 or more during those 100 tosses.

I don't know if that's what you're asking though.

Like FinsRule and MustangSally it is difficult for me to understand how you could be that far off (i.e. 25%). Is it possible we don't understand what you are saying?

Quote: pacomartin

Your math is way off. MustangSally knows how to do this calculation. Out of 100 tosses there is actually a 33.791% probability of observing a streak of 7 or more of either heads or tails. The odds of observing a streak of 3 or more of either heads or tails is nearly 100%.

My math is not way off. Read what I wrote again. For some strange reason, you are talking about the probability of observing 1 or more streaks over 100 flips. I am talking about the average number of streaks over 100 flips, which is clearly what the OP was asking about.

Quote: AxiomOfChoice

I am talking about the average number of streaks over 100 flips, which is clearly what the OP was asking about.

Quote: Bmayo319

Say i am tossing a fair coin 100 times.
The statistics say that I have a 25% chance of observing a streak of 3 or more of either heads or tails.

How is that clear?

removed
silly

Sally

Quote: Bmayo319

Say i am tossing a fair coin 100 times.
The statistics say that I have a 25% chance of observing a streak of 3 or more of either heads or tails.
How much can I count on this probability?
Say a streak of 3 or more only happened 6.25% of the time in my trial, can I calculate how many SDs from mean this outcome is?
Or can I count on seeing a streak of 3 or more being extremely close to 25% over 100 trials?

your answer is 12%. (Don't ask me why or how I got that. just take my word on it).

Did OP ask 3 questions?

Quote: Bmayo319

How much can I count on this probability?
can I calculate how many SDs from mean this outcome is?
can I count on seeing a streak of 3 or more being extremely close to 25% over 100 trials?

3 questions that may have been lost in a translation

Sally

Quote: mustangsally

Did OP ask 3 questions?
3 questions that may have been lost in a translation

Sally

How much can I count on this probability?
12% of the time.

can I calculate how many SDs from mean this outcome is?
Yes

can I count on seeing a streak of 3 or more being extremely close to 25% over 100 trials?
Never or Highly unlikely is a better answer for this question.

Quote: pacomartin

How is that clear?

It's as clear as anything else that he wrote :)

Perhaps "clearly" was an overstatement, but I think that the 25% number gives it away.

Quote: mustangsally

You showed the average number of over-lapping streaks. not too many that tabulate runs do it that way.
I have only seen a few DNA pattern researchers do this and it was only done to show another counting method was accurate.

Well, it's the only way to get an answer of 25%.

Quote: pacomartin

Like FinsRule and MustangSally it is difficult for me to understand how you could be that far off (i.e. 25%). Is it possible we don't understand what you are saying?

Sorry I worded my statement incorrectly, i meant at the beginning of each toss the probability of a 3 or more streak is 25%

Quote: mustangsally

The OP might want to know the expected number of runs of 3 or more in 100 flips of heads or tails
he asked a few different type questions IMO not exactly this one, but I think he would like to know

easily simulated and/or calculated
I show 12.375 calculated and a real fast Excel sim
Average 12.395
SD 2.1943
and the distribution is skewed

Fellers counting method shows
Average 14.049
SD 2.5338
and looks closer to normal than the 3 or more counting method

Sally

Great thanks Sally.

Quote: Bmayo319

Great thanks Sally.

If you choose to flip 3 coins you will have a better 3 Streak average of 14:100 (2% difference but far from 25%)