Fifth example
I have three sticks of the same length. I break the three sticks "completely at random" and take a part of each one at random. What is the probability with which I can construct a triangle that has these three parts as sides?
First example (incorporated in the initial version of the script)
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?
function TruthValue()
{ with(Math) {
/// the current event is: I throw three dice.
/// What is the probability of getting at least 2 equal numbers?
///--------
U1 = floor(random()*6+1);
U2 = floor(random()*6+1);
U3 = floor(random()*6+1);
V=0;
if (U1==U2) { V = 1};
if (U1==U3) { V = 1};
if (U2==U3) { V = 1};
///--------
}}
I have (with "n+": 10000, 10000, 20000, 40000, ...):
n=10240000 44.443642578125% +/- 0.046584667751577526%
n=5120000 44.41736328125% +/- 0.06587676682454288%
n=2560000 44.437421875% +/- 0.09316804416641197%
n=1280000 44.4278125% +/- 0.13175668240289726%
n=640000 44.41859375% +/- 0.18632827992836345%
n=320000 44.4684375% +/- 0.2635377451554643%
n=160000 44.32375% +/- 0.37257684179693506%
n=80000 44.25125% +/- 0.5268164338326546%
n=40000 44.0025% +/- 0.7445942580419068%
n=20000 44.125% +/- 1.0533391879742468%
n=10000 44.23% +/- 1.4900531587301118%
I get an estimate of the probability, which can help me to check the possible
solution found theoretically (in this case I know how to find it:
it is 0.444 ... = 1−5/6*4/6 = 4/9. Why?).
[I reflect. The probability that the 2nd output is different from the first one is 5/6.
The probablity that the third output is different from the first two is 4/6. The
probablity that the 3 outputs are different is 5/6*4/6. So the probability that at
least 2 are equal is:
Second example
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.
function TruthValue()
{ with(Math) {
V = 0; P = 1/8; N = 100; k = 0
for(i=1; i <= N; i++) {if(random() < P) {k = k+1} }
if(k < 11) {V = 1}
}}
n=640000 28.09109375% +/- 0.16854166235983364%
n=320000 28.0340625% +/- 0.2382064168558167%
n=160000 27.92% +/- 0.336455565438695%
n=80000 28.0175% +/- 0.4763291159111611%
n=40000 27.9825% +/- 0.6733780639760818%
n=20000 27.905% +/- 0.9515041811575701%
n=10000 27.61% +/- 1.3412676876377028%
The probability if 28.1%. It is demonstrable (how?) that the probability is 28.09992%.
Third example
The probability (by drawing 10 cards from a 40-pack deck) to get at least a
tris (ie 3 or 4 cards of the same value).
I put in random order the first 40 positive integers in C. I consider the numbers from 1 to 10, ..., from 31 to 40 if they were the cards of the same suit. If C is 37 or 17 ... C%10 is 7. I start from a low value of n.
function TruthValue()
{ with(Math) {
var C = new Array(40); var N = new Array(10);
C[1]=floor(random()*40+1);
for(i=2; i <= 40; i++)
{ U=0; while(U < 1) { C[i]=floor(random()*40+1); U=1; for(j=1; j < i; j++) { if(C[j]==C[i]) {U=0} } } }
V=0; for(i=1; i<=10; i++) {N[i]=0};
i=1; while(i<11) { x=1+C[i]%10; N[x]=N[x]+1; if(N[x]==3) {i=11; V=1} else {i=i+1} }
}}
n=256000 38.399609375% +/- 0.2883747669871522%
n=128000 38.51328125% +/- 0.4080504730615265%
n=64000 38.578125% +/- 0.5772537376074992%
n=32000 39.0125% +/- 0.818041540542717%
n=16000 39.5% +/- 1.1594474524098457%
n=8000 39.575% +/- 1.6402958934906455%
n=4000 38.85% +/- 2.312273994957548%
n=2000 38.7% +/- 3.2681394351654953%
n=1000 37.4% +/- 4.592630210701952%
It is not easy to find the exact value; with the script you can arrive for example to 38.452% +/− 0.129% (or, if you have time, to 38.436% +/− 0.047%).
Fourth example
A matched die gives 1, 2, 3, 4, 5, 6 with, in order, the (rounded) probabilities 0.125, 0.170, 0.240, 0.170, 0.125, 0.170. What is the probability that, if you roll it, you get a multiple of 3?
It is easy to understand that the probability is 0.240 + 0.170 = 0.410. Anyway ...
function TruthValue()
{ with(Math) {
U0 = random();
U=6
if (U0<0.125+0.17+0.24+0.17+0.125) {U=5};
if (U0<0.125+0.17+0.24+0.17) {U=4};
if (U0<0.125+0.17+0.24) {U=3};
if (U0<0.125+0.17) {U=2};
if (U0<0.125) {U=1};
V=0;
if (U==3) { V = 1};
if (U==6) { V = 1};
}}
n=10240000 40.995830078125% +/- 0.04610865907417184% n=5120000 41.00162109375% +/- 0.06520889936050078% n=2560000 41.0290234375% +/- 0.09222870418525478% n=1280000 41.019921875% +/- 0.13042670589904629% n=640000 40.95609375% +/- 0.18440742868955048% n=320000 40.96875% +/- 0.26080402562123445% n=160000 40.9525% +/- 0.3688107637658629% n=80000 41.14375% +/- 0.521947961404103% n=40000 40.8% +/- 0.7372038976698838% n=20000 40.485% +/- 1.041303685011352% n=10000 41.06% +/- 1.4759019576716015%
from which I could deduce that the probability is 40.1% ± 0.05%.
Fifth example | |
function TruthValue()
{ with(Math) {
U1 = random(); U2 = random(); U3 = random();
V=0;
if (U1+U2<U3) { V = 1};
if (U1+U3<U2) { V = 1};
if (U2+U3<U1) { V = 1};
}}
n=81920000 50.00080810546875% +/- 0.01657281528304762%
n=40960000 49.9852685546875% +/- 0.0234374992688422%
n=20480000 49.981855468750005% +/- 0.033145628994871314%
n=10240000 49.997451171875% +/- 0.046875002227913604%
n=5120000 49.974140625000004% +/- 0.06629125834410458%
n=2560000 50.00328125% +/- 0.09375001810867843%
n=1280000 50.028046875% +/- 0.13258255240394223%
n=640000 50.04765625% +/- 0.18750006131752964%
n=320000 50.01625% +/- 0.26516544326225494%
n=160000 50.040625% +/- 0.37500104810078905%
n=80000 50.135% +/- 0.530331467415254%
n=40000 50.18759% +/- 0.7500041016538258%
n=20000 50.105% +/- 1.0606843504617978%
n=10000 49.67% +/- 1.5000423336360444%
If I stop here, I can conclude that the probability is 50.00% ± 0.02%
It can be shown that the probability is exactly 1/2.
Sixth example
I throw a balanced coin 200 times. What is the probability of 100 heads?
function TruthValue()
{ with(Math) {
Head=0
for (var Number=0; Number < 200; Number++)
{
if (random()>0.5) {Head=Head+1;}
}
V=0
if(Head==100) {V=1}
}}
n=2560000 5.662578125% +/- 0.04333616359043924%
n=1280000 5.659296875% +/- 0.06126990857205131%
n=640000 5.64953125% +/- 0.0865784578316723%
n=320000 5.6484375% +/- 0.12242938172716718%
n=160000 5.6175% +/- 0.17269505398482882%
n=80000 5.58125% +/- 0.24348591133078207%
n=40000 5.4875% +/- 0.3416084304459475%
n=20000 5.525% +/- 0.48466505316629543%
n=10000 5.32% +/- 0.6733299171414647%
If I stop here, I can conclude that the probability is 5.66% ± 0.05%
What is the probability of 105 heads?
function TruthValue()
{ with(Math) {
Head=0
for (var Number=0; Number < 200; Number++)
{
if (random()>0.5) {Head=Head+1;}
}
V=0
if(Head==105) {V=1}
}}
n=2560000 4.3852734375% +/- 0.0383938876551426%
n=1280000 4.377109375% +/- 0.054248917055137104%
n=640000 4.378125% +/- 0.07672807688271979%
n=320000 4.29% +/- 0.10746183403954654%
n=160000 4.264375% +/- 0.15153993733268348%
n=80000 4.3275% +/- 0.21581969534460066%
n=40000 4.3675% +/- 0.3065602906836526%
n=20000 4.335% +/- 0.4320044330114068%
n=10000 4.44% +/- 0.6179773050808254%
If I stop here, I can conclude that the probability is 4.385% ± 0.040%
Seventh example
I have to answer 3 questions, each of which accompanied by 4 answers, of which only one is the correct one. If I answer completely randomly, what is the probability that:
A) I answer all the questions exactly;
B) I got one answer wrong;
C) I answer exactly at least two of them ?
I modify TruthValue as shown below, assuming that the OK answer is always the first (r1, r2 and r3 can be 0, 1, 2 or 3).
function TruthValue()
{ with(Math) {
r1=floor(random()*4); r2=floor(random()*4); r3=floor(random()*4)
V=0; if(r1+r2+r3==0) {V=1}
}}
n=20480000 1.5623486328125% +/- 0.008221023443472584%
n=10240000 1.561591796875% +/- 0.011623511463137245%
n=5120000 1.55830078125% +/- 0.016421072238068206%
n=2560000 1.5595703125% +/- 0.023232213332100415%
n=1280000 1.552265625% +/- 0.03277949983245769%
n=640000 1.5453125% +/- 0.0462549232686036%
n=320000 1.531875% +/- 0.06513380394598758%
n=160000 1.54625% +/- 0.09253768021932202%
n=80000 1.5075% +/- 0.1292436506018391%
n=40000 1.485% +/- 0.1814308333874995%
n=20000 1.49% +/- 0.25701022880983865%
n=10000 1.68% +/- 0.3855835933630308%
r1=floor(random()*4); r2=floor(random()*4); r3=floor(random()*4); V=0
if(r1==0 & r2==0 & r3 > 0) {V=1}
if(r1==0 & r3==0 & r2 > 0) {V=1}
if(r2==0 & r3==0 & r1 > 0) {V=1}
n=20480000 14.06828125% +/- 0.023049078094169353%
...
n=10000 13.99% +/- 1.0407017901500901%
r1=floor(random()*4); r2=floor(random()*4); r3=floor(random()*4); V=0
if(r1==0 & r2==0) {V=1}
if(r1==0 & r3==0) {V=1}
if(r2==0 & r3==0) {V=1}
n=20480000 15.614443359374999% +/- 0.024063212276052407%
...
n=10000 15.55% +/- 1.0871963712204706%
A: (1.562±0.009)%; B: (14.07±0.03)%; C: (15.61±0.03)%