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Principle of Symmetry (Quick Note)

Pick n random points on the line (0, 1). This creates n+1 segments on the line. What is the distribution of length of the k-th segment?

Journey:
1. Try solving for the first segment.
2. Argue that the last segment has the same distribution as the first segment, by symmetry.
3. Surprise: the distribution is the same for all segments. Way to see this: instead of line, think of a circle, and argue by symmetry.


The interesting things to note:
Sometimes we can solve for simpler / extremal case, and then extend it to other cases by symmetry argument.


Application
How many cards on average need to be drawn from the deck to get an Ace?

The short answer:
Mark the top of the deck with a Joker card and arrange them in a circle. The 4As + Joker forms 5 segments on the circle. By symmetry, the length of the segments follow the same distribution and have the same average. The averages must add up to 53. So answer is 53/5.

The long answer, if we don't realise the symmetry:
Let F(t) be the probability of having to draw at least t cards from the deck before getting an Ace. Then F(t) = C(53-t, 4) / C(52, 4).

Let f(t) be the probability of having to draw exactly t cards. Then f(t) = F(t) - F(t+1).

Add t*f(t) up and telescoping we will get the answer C(53, 5)/C(52, 4) = 53/5.


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