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Sample spaces and the axioms

Kolmogorov's axioms, events as sets, conditional probability and Bayes.

Probability is a measure on events: non-negative, total 1, additive over disjoint events. Conditional probability P(A|B) = P(A ∩ B)/P(B), and Bayes reverses the conditioning. Picture it: a tree diagram; multiply along branches, add across leaves. Think it: the axioms are measure theory with total mass 1, which is why the two subjects merged.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 1 - (5/6)^4

Arithmetic · Arithmetic

Evaluate 1 - (5/6)^4

1 - \left(\frac{5}{6}\right)^{4}

Àwọn ìṣàmúlò-ètò

1 - \left(\frac{5}{6}\right)^{4} = 1 - \frac{625}{1296}
Power: (5/6)^4 = 625/1296.
1 - \frac{625}{1296} = \frac{671}{1296}
Simplify.

Fi àwọn àgbèwọlé hàn

\frac{671}{1296} \approx 0.51775

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Bii o ṣe le: Sample spaces and the axioms

Power: (5/6)^4 = 625/1296.
Simplify.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Probability Theory

Random variables and expectation The common distributions The law of large numbers and the central limit theorem