maths.free › Probability Theory › 10. Probability › Discussion

Discussion

Bob was late for morning coffee and the group was well into dissecting today's applied combinatorics class.

Discussion

Bob was late for morning coffee and the group was well into dissecting today's applied combinatorics class. As he approached the table, he blurted out Ok guys, here's a problem that doesn't make any sense to me, except that Nadja, my friend from biology, says that if I have a good feel for probability, then it is transparent. Alice not very softly interjected Not much goes through six inches of iron. Bob didn't bite A guy eats lunch at the same diner every day. After lunch, the waiter asks if he wants dessert. He asks for the choices and the waiter replies We have three kinds of pie: apple, cherry and pecan. Then the guy always says I'll have pecan pie. This goes on for six months. Then one day, the waiter says I have bad news. Today, we don't have any apple pie, so your only choices are cherry and pecan. Now the guy says In this case, I'll have the cherry pie. I have to tell you all that this doesn't make any sense to me. Why would the guy ask for cherry pie in preference to pecan pie when he consistently takes pecan pie over both cherry pie and apple pie?

Zori was the first to say something Ok guys, I've finally willing to accept the premise that big integer arithmetic, and things that reflect the same flavor, might and I emphasize might, have some relevance in the real world, but this conversation about dessert in some stupid diner is too much. Xing was hesitant but still offered There's something here. That much I'm sure. Dave said Yeah, a great dessert. Especially the pecan pie. Alice was not amused. All the while Carlos was thinking. Finally, he said I think it has something to do with conditional probability. The patron's preference for pecan pie was conditioned on the fact that there were three choices. When there were only two choices, his preferences changed.

Now Yolanda saw more Doesn't this happen all the time in presidential politics? People prefer candidate \(A\) when \(A\), \(B\) and \(C\) are running, but when candidate \(C\) drops out, they shift their preference to candidate \(B\). Alice said You could say the same thing about close personal relationships. Although she didn't say it, she was thinking that it wouldn't matter how many dropped out if Bob was one of the remaining.

Symbols used here

y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
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.

Spróbuj sam.

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

Więcej w Probability Theory

Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem