We can use simple programs in JavaScript to calculate any probability.
See JavaScript. Some example.
(Copy and paste the lines at the top of this page)
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) {
n=1e4; x=0; for(i=0; i<n; i=i+1) { Head=0; for(j=0; j<200; j=j+1) if(random() > 0.5) Head=Head+1; if(Head==100) 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) { Head=0; for(j=0; j<200; j=j+1) if(random() > 0.5) Head=Head+1; if(Head==100) 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) { Head=0; for(j=0; j<200; j=j+1) if(random() > 0.5) Head=Head+1; if(Head==100) 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) { Head=0; for(j=0; j<200; j=j+1) if(random() > 0.5) Head=Head+1; if(Head==100) 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) { Head=0; for(j=0; j<200; j=j+1) if(random() > 0.5) Head=Head+1; if(Head==100) 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) { Head=0; for(j=0; j<200; j=j+1) if(random() > 0.5) Head=Head+1; if(Head==100) x=x+1 }
document.writeln (x/n*100,"% +/- ",sqrt(x/n*(1-x/n)/sqrt(n-1)*300),"%" )
} </script></pre>
5.54% +/- 0.3962327222918004%
5.84% +/- 0.34154521533431015%
5.525% +/- 0.2798162521554175%
5.6775% +/- 0.2383283524799421%
5.61625% +/- 0.19938989300873827%
5.639375% +/- 0.16799036117620117%