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Discussion

Zori started the conversation with Who in their right mind would trust their lives to an algorithm that used random methods? Xing quickly responded Everyone. At least everyone should.

Discussion

Zori started the conversation with Who in their right mind would trust their lives to an algorithm that used random methods? Xing quickly responded Everyone. At least everyone should. We routinely deal with probabilistic concepts, like getting run over by a bus when crossing the street or having a piano fall on our head. The general public is much more comfortable with notions of probability, even though they may never know the formal definition of a probability space. I for one am completely comfortable taking an airline flight if I can be assured that the probability of a disaster is less than \(10^{-20}\).

Dave wasn't biting on this topic. Instead he offered You have to be struck by the statements that it appears difficult to construct objects which you can prove exist in abundance. I wonder why this is so. Alice said We all find your brain to be a totally random thing, sometimes making sense but often not. There was laughter or at least some snickering. But after a bit, Carlos said There's something fundamental here. Maybe one could prove that there are easily stated theorems which only have long proofs. Bob blurted That doens't make any sense. Zori saw an opportunity where a client would, at considerable expense, commission her to solve a problem (at least better than the competition) that was readily understood but somehow difficult in the end. She knew about the class \(\cgN\cgP\) but maybe there were even bigger challenges (and bigger paychecks) out there.

Symbols used here

n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\bar{x},\ \mu
sample mean, population mean
Average of the data; average of the whole population.
\sigma,\ s,\ \sigma^2
standard deviation, sample s.d., variance
Typical distance from the mean; its square.
P(A),\ P(A \mid B)
probability, conditional probability
Chance of A; chance of A given that B happened.
E[X],\ \operatorname{Var}(X)
expected value, variance
Probability-weighted average of X; its spread.
N(\mu, \sigma^2),\ z
normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.
\mu(A),\ \sigma\text{-algebra}
measure of A
Size of a set; the family of sets that can be measured.

Questions people ask

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

Cubalah sendiri

Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

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