```
We can use simple programs in JavaScript to calculate any probability.
See JavaScript. Some example.

1) In a dice game players throw three dice; who first gets at least 2 equal numbers wins.
What is the probability of getting this in a throw?

<pre><script> with(Math) {
n=1e4; x=0; for(i=0; i<n; i=i+1)
{ U1=floor(random()*6+1); U2=floor(random()*6+1); U3=floor(random()*6+1); s=0
if(U1==U2) s=1; if(U1==U3) s=1; if(U2==U3) s=1;  x=x+s}
document.writeln("n=",n,"  P = ",x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
n=n*10; x=0; for(i=0; i<n; i=i+1)
{ U1=floor(random()*6+1); U2=floor(random()*6+1); U3=floor(random()*6+1); s=0
if(U1==U2) s=1; if(U1==U3) s=1; if(U2==U3) s=1;  x=x+s}
document.writeln("n=",n,"  P = ",x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
n=n*10; x=0; for(i=0; i<n; i=i+1)
{ U1=floor(random()*6+1); U2=floor(random()*6+1); U3=floor(random()*6+1); s=0
if(U1==U2) s=1; if(U1==U3) s=1; if(U2==U3) s=1;  x=x+s}
document.writeln("n=",n,"  P = ",x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
} </script></pre>

n=10000  P = 44.76%  +/- 0.861278010237021%
n=100000  P = 44.536%  +/- 0.4840864325165218%
n=1000000  P = 44.4183%  +/- 0.2721495467311992%

The precision calculated by sqrt(x/n*(1-x/n)/sqrt(n-1)*3 (put after "+/-") is not "certain": there is a
small probability (0.03%) that the deviation of the value found from the true value is greater than it.

2) In a certain population of adults in a given region, a particular childhood disease is known to affect 1 in 8 people.
If you consider 100 adult people from that region, calculate the probability that no more than 10 have been affected by it.

<pre><script> with(Math) {
P = 1/8
n=1e4; x=0; for(i=0; i<n; i=i+1) { k=0; for(j=0; j<100; j=j+1) if(random() < P) k=k+1; if(k<11) x=x+1 }
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
n=n*2; x=0; for(i=0; i<n; i=i+1) { k=0; for(j=0; j<100; j=j+1) if(random() < P) k=k+1; if(k<11) x=x+1 }
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
n=n*2; x=0; for(i=0; i<n; i=i+1) { k=0; for(j=0; j<100; j=j+1) if(random() < P) k=k+1; if(k<11) x=x+1 }
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
n=n*2; x=0; for(i=0; i<n; i=i+1) { k=0; for(j=0; j<100; j=j+1) if(random() < P) k=k+1; if(k<11) x=x+1 }
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
n=n*2; x=0; for(i=0; i<n; i=i+1) { k=0; for(j=0; j<100; j=j+1) if(random() < P) k=k+1; if(k<11) x=x+1 }
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
} </script></pre>

28.36%  +/- 0.780732731440065%
28.095%  +/- 0.6546401245339797%
28.0925%  +/- 0.5504661699402846%
28.19%  +/- 0.4633716818209928%
27.975%  +/- 0.38873889099144%

3) I throw a balanced coin 200 times. What is the probability of 100 heads?

<pre><script> with(Math) {
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
document.writeln (x/n*100,"%  +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )