The discussion centers on calculating the probabilities of flipping two strange coins with adjustable probabilities for heads and tails. For Coin 1, the probability of heads is set at 70% and for Coin 2, it is set at 20%. The calculated probabilities for getting two heads (H&H) is 14% and for two tails (T&T) is 24%, leading to a combined probability of 38% for either event occurring. The conversation also delves into the concept of independence in probability, emphasizing that the outcome of one event does not influence another, and highlights the mathematical intricacies of rolling dice and the improbability of achieving long streaks of the same result.
PREREQUISITESStudents of mathematics, statisticians, game developers, and anyone interested in understanding the complexities of probability and randomness in practical applications.
pwsnafu said:Do you know what independence means?
Fredrik said:Consider a normal six-sided die. The probability to roll a 6 AND then another 6 is ##\frac 1 6\cdot\frac 1 6=\frac{1}{36}##. The probability to roll a 6 OR a 5 is ##\frac 1 6+\frac 1 6=\frac 1 3##. A good rule of thumb is that an "and" means that you need to multiply the probabilities, and an "or" means that you need to add them.
Two events A and B are independent if knowing that one of them occurred doesn't affect the probability that the other one occurred. Suppose that you roll two dice, and see that the first one is a 6. This knowledge doesn't change the probability that the other one is a 6.
My suggestion that you interpret "and" as "multiply the probabilities" is only good when the events are independent.
Fredrik said:Suppose that you roll two dice, and see that the first one is a 6. This knowledge doesn't change the probability that the other one is a 6.
StrangeCoin said:But as the number of rolls increases, doesn't it become increasingly unlikely that you would keep rolling the same number?
Getting 6 ten times in a row is less probable than rolling it only 2-3 times, so doesn't that mean that each roll is not really independent but somehow related to its history and past time?
Looks good. Note that the probabilities of the four possible sequences (HH, HT, TH and TT) add up to 1. It's a good idea to do sanity checks like this when you do probability calculations, because it's so easy to make mistakes.StrangeCoin said:Ok, thanks. Let me try then.
Coin 1 Heads = 70%, Tails = 30%
Coin 2 Heads = 20%, Tails = 80%
---
Chance of H&H = 0.7 * 0.2 = 0.14
Chance of T&T = 0.3 * 0.8 = 0.24
Chance of H&H or T&T = 0.14 + 0.24 = 38%
pwsnafu said:The reason why "getting 6 ten times in a row is less probable than rolling it only 2-3 times" is true is because there are more permutations of the latter. Both 6612345123 and 1234512366 have two sixes, but there is only one way to roll 6666666666.
This is incorrect. The probability is 1/6, regardless of what the previous results were.StrangeCoin said:Yes, but makes this true as well, it still holds: rolling 6 on 10th try is less likely if previous nine rolls were all 6, than if they were not.
Fredrik said:This is incorrect. The probability is 1/6, regardless of what the previous results were.
StrangeCoin said:I can't disagree with that. I just feel there is more to it. Can we test it? I could write a little dice rolling program, just not sure what exactly to test for, or test against, what to compare and how.
StrangeCoin said:we should never roll 6-side 100 times in a row
This is wrong. Let's denote the two sides of a coin by 0 and 1. If you flip a coin 10 times, it's likely (more than 50%) that the sequence of results will contain the subsequence 111. If you flip it 100 times, it's likely that the sequence will contain the subsequence 11111. (I haven't actually calculated those probabilities, so I'm not sure if that the length of the subsequence should be 5, but you get the idea). Think of any subsequence, of any length (for example "a million 1s"). There's always a number N such that if you flip the coin N times, it's likely that your subsequence will appear.StrangeCoin said:I'm talking about distribution, sequences and streaks, rather than just overall or total statistics. What I'm saying is that if we roll a dice for an eternity we should never roll 6-side 100 times in a row. And if it happens, then it means the rolls were not actually random, because that's completely the opposite of what random is supposed to be.
micromass said:In order to roll a one a 100 times in a row, we need to roll the dice approximately ##783,982,348,200,085,087,316,028,320,589,669,384,644,572,452,567,545,845,851,686,359,643,396,569,772,850##
This is a staggering number. It is approximately ##7\times 10^{77}##. The scientific name for such a number is ##783## quattuorvigintillion.
So where is the boundary where you do expect to see ##n## consecutive heads, but suddenly ##n+1## is totally impossible and indicates the dice is loaded? There shouldn't be such a boundary. All rolls of consecutive heads should be possible, but not all should be likely.
I get the impression that you've been confusing two rather different problems. One is the problem of rolling a die multiple times. The other is the weird situation that you've been sent a die from the factory and then been told that half the dice they sent out were defective and always stop on 6. In this problem, each 6 you roll makes it more likely that you are in fact rolling the defective die. But in the other problem, the die is by assumption not defective.StrangeCoin said:Consider if the universe started rolling sixes from the beginning of time, every second another roll and every time it's a six, and so on to this day and into the eternal future. You would say that's just as likely as any other combination of the 6 possibilities, and I'm saying it doesn't even apply, it does not compute because such sequence is not random to start with, it's the opposite.
Fredrik said:I get the impression that you've been confusing two rather different problems. One is the problem of rolling a die multiple times. The other is the weird situation that you've been sent a die from the factory and then been told that half the dice they sent out were defective and always stop on 6. In this problem, each 6 you roll makes it more likely that you are in fact rolling the defective die. But in the other problem, the die is by assumption not defective.
StrangeCoin said:In order to get a sequence of two consecutive sixes we need to roll the dice approximately 42 times. If we don't get two sixes after 42 rolls, every roll after that simply must become more and more likely to yield a six as that's what we observe happens in reality.
micromass said:No! We do not observe this in reality! This is called the gambler's fallacy and is confusing to so many people trying to deal with probability. The chances of getting a six never increase. Not in the mathematical theory as in the observations!
What does the 42 throws mean then? It means that if I do the experiment a lot of times (say 100 times) and if I record the number of throws each time, then the average of these numbers will be 42. This means that sometimes I will get two sixes almost immediately after say 4 rolls. And sometimes I will need 100 rolls.
I have said that the mathematics describes the randomness in nature perfectly. But please, do not take my word for it. Test it yourself! Do a version of the experiment above: "Throw a dice until you have a six. Record the number of throws you needed to do before you got 6".
Do the experiment 50 or 100 times and record the scores. You will see that the different outcomes are widely different. This illustrates the fact that you cannot say anything meaningful about the different individual experiments. But you can say something about the whole! Indeed, take the average of the scores you got and you will see that you got very close to 6.
Then, if you're good with programming. Do the above experiment but only record the number of throws until you got 5 consecutive sixes. Then do this experiment 1000 times with the computer. Look at the different outcomes, they will look very different, but the average should lie around 9330.
StrangeCoin said:I think you are basically saying anything and everything is possible given enough time, for you there is no "impossible", but instead you have "highly unlikely". On the other hand I'm saying some things are just downright clear cut impossible, like 100% impossible, in this universe as it is.
I'm not disagreeing with you. I'm just saying that if you ever really don't get a single 6 in 100 rolls it would be very unnatural, and just as likely to cause the universe to disappear and be replaced by something else.
The problem is I can prove my claim, but you can not prove yours. I can say you will never roll a dice 100 times without getting a single six. If you attempt you will most likely, if not certainly, be wasting time proving my claim instead. The question is only how low I can go, where is it that 'highly unlikely' becomes 'truly impossible'. Maybe 30 rolls without a single six is already 100% impossible. Without practically confirming otherwise you can't really be sure, so you must let it be a possibility too.
I don't think 5 in a row is enough. To reach "impossible" I suspect we need at least 30 of a kind in a row, or 30 rolls without getting some number at all.
#lang racket
(require math)
(define distr (discrete-dist '(0 1) '(1 5)))
(define (random-dice-roll)
(car (sample distr 1)))
(define (Experiment n)
(define (iter counter number_throws)
(define d (random-dice-roll))
(cond [(= counter n) (fprintf (current-output-port) "~a" number_throws)]
[(= d 1) (iter (+ counter 1) (+ number_throws 1))]
[(= d 0) (iter 0 (+ number_throws 1))]))
(iter 0 0))
(Experiment 100)micromass said:The computer ran for quite a long time, 10 or 20 minutes. Finally, it did tell me that there has been a consecutive sequence of 100 1's. The computer also told me that it took ##21,580,227## dice rolls for such a sequence to arise.
#lang racket
(require math)
(define distr (discrete-dist '(1 2 3 4 5 6) '(1 1 1 1 1 1)))
(define (random-dice-roll)
(car (sample distr 1)))
(define MyPort (open-output-file "test.txt" #:exists 'append))
(define (Experiment n)
(define (iter counter number_throws)
(define d (random-dice-roll))
(fprintf MyPort "~a " d)
(cond [(= counter n) (fprintf (current-output-port) "~a" number_throws)]
[(not (= d 6)) (iter (+ counter 1) (+ number_throws 1))]
[else (iter 0 (+ number_throws 1))]))
(iter 0 0))
(Experiment 100)
(close-output-port MyPort)micromass said:I have modified the code such that it saves all the dice rolls to a (possibly huge) file.
6: ++Longest streak= 2
34: ++Longest streak= 3
45: ++Longest streak= 4
1,989: ++Longest streak= 5
6,603: ++Longest streak= 6
7,061: ++Longest streak= 7
542,606: ++Longest streak= 8
3,719,614: ++Longest streak= 9
25,282,514: ++Longest streak= 10
25,282,515: ++Longest streak= 11
100,800,688: ++Longest streak= 12
203,506,206: ++Longest streak= 13
4,294,967,295: DONE!StrangeCoin said:I wouldn't slow it down with that. Anyway, there must be something wrong with either your program or RNG function. This is what I get:
Code:6: ++Longest streak= 2 34: ++Longest streak= 3 45: ++Longest streak= 4 1,989: ++Longest streak= 5 6,603: ++Longest streak= 6 7,061: ++Longest streak= 7 542,606: ++Longest streak= 8 3,719,614: ++Longest streak= 9 25,282,514: ++Longest streak= 10 25,282,515: ++Longest streak= 11 100,800,688: ++Longest streak= 12 203,506,206: ++Longest streak= 13 4,294,967,295: DONE!
My function can pick any of the six numbers at random and then it compares that current roll with the previous roll. If they are the same I increase "cur_streak" counter, and then if "cur_streak" is greater than "max_streak" counter, the "max_streak" gets the value of "cur_streak and the information is printed as shown above. Any streak is valid, sixes, fives, fours, threes, twos and ones, just the same.
It goes pretty "quickly" to 10-11 streak, then it takes some time to reach 12-13, and after that it just stays there, for like, ever. You can see the curve bends pretty sharply, it's obviously converging toward infinity, to "infinitely unlikely" or "impossible". I bet it wouldn't go over 15 streaks even I leave it to run for weeks, and possible never.
I think you would actually have a better luck with some more "natural" source of randomness than computer RNG algorithms, which by design might be doomed to never produce more than, say 15 streaks.
6: ++Longest streak= 2 Longest streak predicted by theory: 1
34: ++Longest streak= 3 Longest streak predicted by theory: 2
45: ++Longest streak= 4 Longest streak predicted by theory: 2
1,989: ++Longest streak= 5 Longest streak predicted by theory: 4
6,603: ++Longest streak= 6 Longest streak predicted by theory: 5
7,061: ++Longest streak= 7 Longest streak predicted by theory: 6
542,606: ++Longest streak= 8 Longest streak predicted by theory: 7
3,719,614: ++Longest streak= 9 Longest streak predicted by theory: 8
25,282,514: ++Longest streak= 10 Longest streak predicted by theory: 9
25,282,515: ++Longest streak= 11 Longest streak predicted by theory: 9
100,800,688: ++Longest streak= 12 Longest streak predicted by theory: 10
203,506,206: ++Longest streak= 13 Longest streak predicted by theory: 10
4,294,967,295: DONE! Longest streak predicted by theory: 12micromass said:But let's see. What if you want a streak of 14, how much trials should you let your computer do? The theory says to run it for 15,672,832,819 iterations, which is three times as long as you have done know. What about a streak of 15? You should run it for 94,036,996,915 which is about 19 times as long as you have done now.
Also, random generated numbers are not designed to never produce 15 consecutive numbers.
So since there is a reasonable agreement between your table and my theoretical values, why do you expect this agreement to break down eventually? It makes no sense.
What if you want to get over 15, which you say is impossible? The theory says to let it run 564,221,981,491 iterations which is about 100 times as long as you done now. So if you have the memory and the time, please let it run for a couple of days, you will go over 15!
3: ++Longest streak= 2
22: ++Longest streak= 2
26: ++Longest streak= 2
27: ++Longest streak= 3
63: ++Longest streak= 3
88: ++Longest streak= 3
135: ++Longest streak= 3
170: ++Longest streak= 3
190: ++Longest streak= 3
333: ++Longest streak= 3
348: ++Longest streak= 3
375: ++Longest streak= 3
463: ++Longest streak= 3
471: ++Longest streak= 3
514: ++Longest streak= 3
530: ++Longest streak= 3
531: ++Longest streak= 4
626: ++Longest streak= 4
713: ++Longest streak= 4
754: ++Longest streak= 4
755: ++Longest streak= 5
756: ++Longest streak= 6
757: ++Longest streak= 7
758: ++Longest streak= 8
266,742: ++Longest streak= 8
266,743: ++Longest streak= 9
266,744: ++Longest streak= 10
13,849,721: ++Longest streak= 10
60,694,700: ++Longest streak= 10
60,694,701: ++Longest streak= 11
90,048,622: ++Longest streak= 11
126,461,427: ++Longest streak= 11
128,375,177: ++Longest streak= 11
128,375,178: ++Longest streak= 12
662,680,498: ++Longest streak= 12
1,395,635,122: ++Longest streak= 12
1,547,176,046: ++Longest streak= 12
1,738,987,914: ++Longest streak= 12
1,769,359,311: ++Longest streak= 12
1,907,939,721: ++Longest streak= 12
2,010,645,238: ++Longest streak= 12
2,010,645,239: ++Longest streak= 13
4,158,128,887: ++Longest streak= 13
4,294,967,295: DONE!StrangeCoin said:I've already tested for well over 15,672,832,819 rolls, by repeating the test. It never went above 13 in a row streak. I will upgrade roll counter from unsigned long to unsigned long long so I can run up to 18,446,744,073,709,551,615 rolls in one go.
Just a feeling.
micromass said:OK, but then the discussion is over. I told you about the theory. I showed you some results how the theory matched with the results, both my results and your results. If you're going to discard this based on a feeling, then all my efforts are useless anyway. I can spend my time better elsewhere.
#include <time.h>
#include <stdio.h>
void printfcomma (unsigned long long n) {
if (n < 1000) {
printf ("%llu", n);
return;
}
printfcomma (n/1000);
printf (",%03llu", n%1000);
}
void main()
{
system("cls");
srand(time(NULL));
int cur_streak= 1;
int max_streak= 1;
int last, dice;
unsigned long long rolls= 0;
do
{
dice= rand()%6;
if(dice == last)
{
cur_streak++;
if(cur_streak > max_streak)
{
max_streak = cur_streak;
printfcomma(rolls);
printf(": ++Lucky Streak = %d \n", max_streak);
}
}
else
cur_streak= 1;
rolls++;
last= dice;
}while(rolls < 94036996915); // 94,036,996,915 testing now, 564,221,981,491 later
printfcomma(rolls);
printf(": DONE!\n");
}