If a biased coin flips heads with probability $\frac{3}{7}$ and tails with probability $\frac{4}{7}$ is flipped $80$ times, what is the probability that heads is flipped $30$ times?
I used binomial distribution so I get $$ {80 \choose 30}\left(\frac{3}{7}\right)^{30}\left(\frac{4}{7}\right)^{50}\\ =\frac{80!}{30!50!}\left(\frac{3}{7}\right)^{30}\left(\frac{4}{7}\right)^{50} $$
Is this right?
I agree with above comment.
Of course I presume that you should add that they are independently distributed. This is correct then.
And that you not talking about 30 heads in a row, or a specific 30 heads sequence? This is correct
As you will notice that the Probability value ~$0.057$ is close to the maximum probability for getting $n$ heads in $80$ tosses which occurs at about $n=34$ (roughly $3/7 \times 80$) which is generally the most likely frequency value, specific value for IID trials, when one is considering all possible combinations that could lead to that relative frequency and not a singular sequence or specific.
This is insofar as the relative frequency is closest to the probability values permutation.
Asymptotically speaking, the probability of getting that exact value frequency=probability value, lessens (particular if p=0.5) as $n$ grows to infinity, but the probability of getting approximately that relative frequency value, increases, and converges toward $1$, and so this is why as n goes to infinity one, we say that the probability of getting that relative frequency value is one.
In our real numbered system, relative frequency values infinitesimally close to $0.5$ are treated as $0.5$ if the deviation frequency, limits to zero (and treated as zero) and cannot be expressed by a real number
for example : $$\frac{[0.5 \times n] + 10]}{n}\,\text{ as}\, n \to \infty\, \text{will be treated as } = 0.5$$
Incidentally, In non standard analysis, however, this is often made more rigorous
As the deviation from the frequency might need to be un-countably smaller, so that it does not qualify as a infinitesimal non zero quantity, otherwise, the relative frequency value may qualify as distinct (in value) from $0.5$.
$[0.5+ \delta] \neq 0.5$, where $\delta$ is a infinitesimal amount\, $\delta \neq 0$ in non-standard analysis)
Otherwise
$$\text{Otherwise, in the strong law of large numbers, both the probability} \,1\,\text{term, may be replaced with}\, '(1- \epsilon)\, \text{ and the term}' \, \text{|prob-relfreq|', qualified by}\, '<\epsilon'\,\text{in non-standard analysis approaches to probability and measure theory}$$.
As it may not qualify as getting that specific relative frequency value unless one gets that specific exact frequency value as infinite quantities are treated akin to finite quantities in non standard approaches (so that the probability of getting that exact limiting relative frequency = probability in IID trials would be rather low, (in non standard analysis) .
Whereas one might not say in standard analysis, but qualify this instead as that 'exact frequency' where differing frequencies qualify as the same relative frequency .
Unlike perhaps in non standard analysis.