Foundation lesson
A probability, a frequency and a single trial
A model probability counts the favoured outcomes among equally likely ones. A frequency counts what happened over many trials. A single trial either happens or does not. The three answer different questions.
Written for this edition, not translated from Einstein. Editorial review pending.
What does a probability of 1/4 say about one try, and about a hundred?
Drop two tokens into a box, each landing in the left or the right half with equal chance. There are four outcomes: left and left, left and right, right and left, right and right. One of the four is both left, so the model probability of both left is 1/4.
Now drop them 100 times and count. Both might land left 23 times: a frequency of 23/100. Over more and more trials the frequency tends to settle near 1/4, but any particular count can differ from 25.
A single drop either lands both left or it does not. A probability of 1/4 is not a quarter of an outcome. It says how often, over many drops.
The light-quanta paper counts in exactly this way in §5. The chance that n independent points are all in a part v of a volume v₀ is (v/v₀)ⁿ, and the paper calls it a statistical probability: a frequency over many moments.
Worked example: Counting outcomes for two tokens and for four
- Two tokens: 4 equally likely outcomes, 1 of them both left, so the probability is 1/4.
- Four tokens: 2⁴ = 16 outcomes, 1 of them all left, so the probability is 1/16, which is (½)⁴.
- A run of 100 drops of four tokens might show all left 5 times: a frequency of 5/100, near 1/16 (about 6 in 100) but not equal to it.
- Any one drop shows all left or it does not. The 1/16 is about the long run.
Where this lesson stops
A probability counts favoured outcomes among equally likely ones; a frequency counts what happened. How a probability becomes an entropy is the lesson on entropy and the number of ways.
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