Philosophy Now Forum

Agent Smith, I have a grudging respect for your posts. So edify me. What's the paradox?

alan1000 wrote: ↑Wed Nov 30, 2022 4:19 pm So how do you explain the fact that, given a sufficiently large sample, the mean tends to approach closer and closer to a true average?

Lets try this another (to make you realise that you have some hidden assumptions) way shall we?

Suppose the set of natural numbers ℕ assuming an infinite sample size what is its "true average" which the mean approaches?

alan1000 wrote: ↑Wed Nov 30, 2022 4:19 pm So how do you explain the fact that, given a sufficiently large sample, the mean tends to approach closer and closer to a true average?

alan1000, you may find this to be of interest.

https://en.wikipedia.org/wiki/Law_of_large_numbers

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.

You can also google around for "law of large numbers" to find many other references.

alan1000 wrote: ↑Wed Oct 05, 2022 2:58 pm It's often said that "the dice have no memory". In ten coin tosses, if the first 9 come up heads, it doesn't make it any more likely that the tenth throw will come up tails; the probability remains 50% or, as the statisticians prefer to say, 0.5.

This is provably true for a limited sample of 10 tosses. But does it remain true for an infinite number of tosses?

Opinion polls usually try to get at least 1000 responses because they know that the more responses you get, the more reliable the conclusions you can draw from it. Given ever-larger samples, the true average tends to assert itself.

But what 'you', people, THINK or SAY does NOT really have much to do with how a coin lands.

alan1000 wrote: ↑Wed Oct 05, 2022 2:58 pm So, let us suppose the gambler intends to remain in the game for a (theoretically) infinite number of throws. In a million tosses of the coin, all things being equal, we would expect that half would come up heads and half tails. So if the last 10 have come up heads, we would expect that at some future stage this would be balanced with a series of 10 tails.

But this is ONLY if one was ASSUMING some 'thing' to be true.

alan1000 wrote: ↑Wed Oct 05, 2022 2:58 pm And the future begins with the next toss.

The 'future' ALWAYS BEGINS HERE-NOW. Just like the 'past' ALWAYS ENDS HERE-NOW.

But anyway;

alan1000 wrote: ↑Wed Oct 05, 2022 2:58 pm If the gambler a) intends to stay in the game for an infinite period, and b) is reasoning inductively, then, it would be wise to assume that if the last 9 throws have come up heads, the probability is increased that the next throw will be tails.

BUT, ASSUMING some 'thing' to be true, when NOT KNOWING what IS ACTUALLY True, is NOT REALLY that 'wise' AT ALL.

alan1000 wrote: ↑Wed Oct 05, 2022 2:58 pm Is this a reasonable assumption?

'Reasonable' in relation to 'what', EXACTLY?

If the "gambler" has ALL of its, and its family's, money on that one coin toss coming up 'heads', then would this be 'reasonable', to 'you' "alan1000"?

And, if 'you' say, 'Yes', then would that have been 'reasonable' to the "gambler's" wife, for example?

Would that "gambler" be ABLE to REASON WHY it just LOST ALL of THEIR money, on a coin toss?

If 'Yes', then REALLY?

But, if 'No', then it was OBVIOUSLY NOT a 'reason-able' ASSUMPTION, AT ALL.

But, then again, 'you' might have ANOTHER scenario, out of the COUNTLESS OTHER scenarios, that 'you' would use here.