Grades Algebra II & Trigonometry Probability & StatisticsExpected Value & Bayes' Theorem

Expected Value & Bayes' Theorem

🎲 Probability & Statistics

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Expected Value & Bayes' Theorem

Compute expected outcomes and update probabilities with new evidence.

Expected value E(X) is the long-run average outcome of a random variable — the probability-weighted average of all possible values. Bayes' Theorem provides a framework to update the probability of a hypothesis given new evidence.

E(X) = \sum_{i} x_i \cdot P(X = x_i)

P(A \mid B) = \frac{P(B \mid A)\cdot P(A)}{P(B)} = \frac{P(B \mid A)\cdot P(A)}{P(B \mid A)P(A) + P(B \mid A^c)P(A^c)}

🎲Expected Value of a Die Roll

For a fair six-sided die: E(X) = 1·(1/6) + 2·(1/6) + 3·(1/6) + 4·(1/6) + 5·(1/6) + 6·(1/6) = 21/6 = 3.5.

E(X) = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5

🩺Bayes' Theorem — Medical Test

Disease prevalence: 1%. Test sensitivity: 90%. Specificity: 95%. P(disease|positive) = (0.9·0.01)/[(0.9·0.01)+(0.05·0.99)] = 0.009/0.0585 ≈ 15.4%. Even a positive test may not mean disease!

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Remember This!

A game is "fair" if E(X) = 0 (you expect to neither gain nor lose in the long run). If E(X) < 0, the house has the advantage. Always compute expected value before gambling!

✏️ Try It!

A game pays $5 with probability 0.3 and costs $2 with probability 0.7. What is the expected value?

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