Bayes Theorem Explained with Venn Diagrams and Proof

Описание: 🔑 Bayes Theorem | Conditional Probability | Probability Theory | Set Theory

🔵 The Topic: Bayes theorem explained with visual proof using Venn diagrams.

📐 The Explanation (Step-by-Step):
This video explains Bayes theorem using set theory and Venn diagrams. The presentation begins with two non-empty sets A and B within a universal set U, shown as overlapping circles in maroon and cyan colors with their intersection A∩B displayed in blue. The probability of A given B is defined as P(A|B) = |A∩B|/|B|, where the vertical bars represent cardinality (the number of elements in a set). Similarly, the probability of B given A is P(B|A) = |A∩B|/|A|.

The video then multiplies both expressions by 1/|U| divided by 1/|U|, which transforms the cardinality ratios into probability notation. This gives P(B|A) = P(A∩B)/P(A) and P(A|B) = P(A∩B)/P(B). Since both expressions contain P(A∩B), the video shows how to combine these equations. First, it expresses P(A∩B) = P(B|A)·P(A), then substitutes this into the equation for P(A|B). The final result is the Bayes theorem formula: P(A|B) = [P(B|A)·P(A)]/P(B), which is highlighted in a white box at the end.

The visual presentation features a black background with a yellow-bordered rectangle containing the Venn diagram. The formulas are displayed in white and colored text, progressing from cardinality notation to probability notation, clearly showing each algebraic step in the derivation.

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