# Thread: Please help solve an argument

1. ## Please help solve an argument

Hello,

My self and a friend disagree about a probability question.

He thinks that if a truly random dice is thrown an infinite number of times, it
is still possible to never throw a 6. No single throw guarantees a 6, so an
infinite number of single throws does not garuentee a six.

I, however, an inclined to disagree. I suspect that the 1/6 probability of
throwing a 6 on a truly random dice becomes not a probability, but a
necessary ratio when the dice is thrown an infinite number of times.

Neither of us are mathematicians and we are both humble in our hunches.

Can anyone tell me who is right and show the reason/proof that each
person is right/wrong? I would be very grateful.

ed: Please only answer in very simple maths or, even batter, words if
this is possible as neither of us are mathematicians at all.

I apologize if I have the wrong forum or have made some other mistake.

Thankyou

2. In the first place it is impossible to throw a die an infinite number of times, but if you throw it any finite number of times it is always possible you will never yet have thrown a 6. The die does not know it is supposed to throw a 6 so if you throw it 100 times and never get a 6, you are no more likely to get a 6 on the next throw than you were on the very first throw. So your friend is basically right. On the other hand it is very improbable you would even throw the die more than 64 times without getting a six - the probability of throwing 65 consecutive non sixes is (5/6)^65 which is about 1 in 1 million. If you throw the die once per second you could expect to get 65 consecutive non sixes somewhere between about once every 10 days and once a year (I haven't calculated it exactly - 10 days is too low, but is about the time it would take to throw the dice a million times, and a year is the time it might take to make about a million independent trials of throwing the dice 65 times). But now it would take about 1 million times that long to expect to get 128 consecutive non-sixes, so unless you had about a million people all throwing dice continuously you would probably not live long enough for it to happen.

3. Originally Posted by Letum
Hello,

My self and a friend disagree about a probability question.

He thinks that if a truly random dice is thrown an infinite number of times, it
is still possible to never throw a 6. No single throw guarantees a 6, so an
infinite number of single throws does not garuentee a six.

I, however, an inclined to disagree. I suspect that the 1/6 probability of
throwing a 6 on a truly random dice becomes not a probability, but a
necessary ratio when the dice is thrown an infinite number of times.

Neither of us are mathematicians and we are both humble in our hunches.

Can anyone tell me who is right and show the reason/proof that each
person is right/wrong? I would be very grateful.

ed: Please only answer in very simple maths or, even batter, words if
this is possible as neither of us are mathematicians at all.

I apologize if I have the wrong forum or have made some other mistake.

Thankyou
It depends on what you (and your friend) mean by "impossible". If by event A, say, being "impossible" you mean that the probability of its occurrence is $\displaystyle 0$, then you are right. (I suppose the occurrence of an event that has probability 0 would rightly be called "a miracle". So discounting the occurrence of miracles: you are right.)

If you (and your friend) mean by "impossible" that there is something in the very structure of the random experiment that prevents such an event from occurring (like a sequence of throws that has a 7 in it ), then your friend is right.

Within the theoretical framework of probability theory, insofar as it is applicable to infinite(!) sequences of throws, you may safely assume that you are right. Note that an infinite sequence of throws of an actual dice are out of the question. Note also, that for finite sequences of throws of a fixed length n the probability of being without a 6 in it is $\displaystyle >0$, and in that sense this is possible (even quite likely if n is small, namely $\displaystyle \left(\tfrac{5}{6}\right)^n$). Since infinite sequences of throws cannot be achieved by experiment, we don't actually know whether our world is structured that way that an infinite sequence without a single 6 is possible or not. Maybe there is some kind of demon (or god) that interferes as soon as one reaches a sequence of length $\displaystyle >10^{40}$ without a 6 in it, by arbitrarily preventing any throws of 6 to occur after $\displaystyle 10^{40}+1$ throws. If such a demon (or god) existed, your friend would surely be right.
But note: If there were such a demon, then an infinite(!) sequence of throws of actual dice would not be correctly modelled by an infinite sequence of independend throws of (theoretically ideal) dice. So this would be a case where the theoretical model simply does not apply for $\displaystyle n>10^{40}$.

4. Thankyou Aluwn.
I'm aware that the probability of throwing a dice missions of times and not
getting at least one six is minute and that previous throws do no effect future throws.
The more times the dice is thrown, the closer to 0 the probability that a 6 has
not been thrown, but for all finite number of throws the probability remains
above 0, however small the number. I would like to know if it becomes 0 for
a non-finite number of throws.

I am also aware that it is impossible to actually throw a dice an infinite
number of times, but I don't think that is relevant. Surely maths can answer
questions about simple data sets that are impossible to collect(?).

Besides, it is theoretically possible to come across a data set containing
information about an infinite number of throws; should such a data set exist
in the universe.

Thankyou Failure.

It depends on what you (and your friend) mean by "impossible"
I am not sure what you mean when you say this because I did not
use the word "impossible" in my post.

Since infinite sequences of throws cannot be achieved by experiment, we don't actually know whether our world is structured that way that an infinite sequence without a single 6 is possible or not
I was under the impression that maths did not rely on experiment for
proofs. Perhaps I am wrong; I am humble in my understanding of maths.

5. Originally Posted by Letum
Thankyou Failure.

I am not sure what you mean when you say this because I did not
use the word "impossible" in my post.
Agreed. But does it matter if you define impossible to mean not possible? Becaues if you do that you can define either "possible" or "impossible" and get the other concept by simple negation.

I was under the impression that maths did not rely on experiment for
proofs. Perhaps I am wrong; I am humble in my understanding of maths.
Agreed. Math in itself does not depend on experiment. However, the question is whether the mathematical model of infinite sequences of independent throws of ideal dice applies to the throwing of actual dice. And this latter is a question that might require experimentation. For example, elementary particle physics tells us that there are some objects that behave quite differently than our theoretical dice, coins and so on, as regards probabilities. Probability theory in itself would not have taught us any such thing. It was experiments that required replacing well worn (and to us humans highly plausible) mathematical models by other (equally mathematical but now to us humans not at all particularly plausible) models that deliver probabilities that are in agreement with experimental data.

6. Ah, I think I see what you mean.
I only wish to talk about the ideal, theoretical dice; a Platonic ideal of a dice
that consistently produces one of 6 numbers with a probability of 1/6 for each
number, regardless of anything else.