*assuming a fair coin, getting 10 top in a row whilst tossing a coin walk not rise the chance of the next coin toss being a tail*, no matter what lot of probability and/or statistical jargon is tossed approximately (excuse the puns).

Assuming that is the case, my question is this: how the hell execute I convince someone that is the case?

They are smart and educated yet seem determined not to think about that I could be in the appropriate on this (argument).

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they are trying come assert the <...> if there have been 10 heads, climate the following in the succession will much more likely be a tail because statistics states it will counter in the end

There"s only a "balancing out" in a very certain sense.

If it"s a fair coin, then it"s quiet 50-50 in ~ every toss. The coin *cannot recognize its past*. The cannot understand there was an overabundance of heads. It cannot compensate for its past. *Ever*. It simply goes on randomly being heads or tails with continuous chance the a head.

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If $n_H$ is the number of heads in $n=n_H+n_T$ tosses ($n_T$ is the variety of tails), for a fair coin, $n_H/n_T$ will have tendency to 1, as $n_H+n_T$ goes to infinity .... However $|n_H-n_T|$ doesn"t walk to 0. In fact, it *also* goes come infinity!

That is, nothing plot to do them an ext even. The counts *don"t* have tendency toward "balancing out". On average, imbalance between the count of heads and also tails in reality grows!

Here"s the an outcome of 100 set of 1000 tosses, v the grey traces mirroring the distinction in number of head minus variety of tails at every step.

The grey traces (representing $n_H-n_T$) room a Bernoulli random walk. If friend think of a particle moving up or under the y-axis through a unit action (randomly with equal probability) at every time-step, then the circulation of the position of the fragment will "diffuse" away from 0 end time. That still has 0 supposed value, yet its meant distance native 0 grows together the square source of the variety of time steps.

The blue curve above is at $\pm \sqrtn$ and also the environment-friendly curve is at $\pm 2\sqrtn$. As you see, the typical distance between total heads and also total tails grows. If there was anything exhilaration to "restore to equality" - to "make increase for" deviations indigenous equality - castle wouldn"t have tendency to frequently grow further apart favor that. (It"s not hard to present this algebraically, yet I doubt that would certainly convince your friend. The an important part is that the variance the a sum of independent random variables is the amount of the variances $$ -- every time you include another coin flip, you include a constant amount top top the variance of the sum... For this reason variance must thrive proportionally v $n$. In turn the conventional deviation boosts with $\sqrtn$. The continuous that gets added to variance in ~ each action in this situation happens to be 1, yet that"s not critical to the argument.)

Equivalently, $\fracn_H-n_Tn_H+n_T$ does go to $0$ together the total tosses goes to infinity, however only because $n_H+n_T$ goes come infinity a lot quicker than $|n_H-n_T|$ does.

That method if we divide that cumulative count *by $n$* at every step, it curves in -- the usual absolute distinction in counting is of the order of $\sqrtn$, however the typical absolute distinction in *proportion* need to then it is in of the bespeak of $1/\sqrtn$.

That"s every that"s going on. The increasingly-large* arbitrarily deviations indigenous equality are simply "*washed out*" by the *even bigger* denominator.

* raising in common absolute size

See the little animation in the margin, here

If your friend is unconvinced, toss some coins. Every time you gain say 3 heads in a row, obtain him or she to nominate a probability because that a head on the next toss (that"s less than 50%) the he thinks should be fair by his reasoning. Ask for them to give you the matching odds (that is, he or she need to be ready to salary a bit much more than 1:1 if you gambling on heads, because they insist that tails is an ext likely). It"s finest if it"s set up as a many bets each for a small amount of money. (Don"t it is in surprised if there"s some excuse as to why they can"t take it up their fifty percent of the gambling -- yet it does at least seem to dramatically reduce the vehemence with which the position is held.)