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Probability (Standard)

Multi-step exam-style items.

Phase 2 · Gaokao

FoundationsStandardAdvanced

Probability: core idea

Independent events multiply.

Example problem: hint and answer

Conditional from: P(A∩B)=0.1P(A\cap B)=0.1, P(A)=0.5P(A)=0.5: P(B∣A)P(B|A)?

Hint: Ratio. Answer: 0.2.

Practice (9 problems — type your answers)

  • 1. Binomial n=3n=3, p=1/2p=1/2: P(X=2)P(X=2)?

  • 2. Expectation of fair die?

  • 3. Var Bernoulli p=0.4p=0.4?

  • 4. Without replacement 3 red of 10: P(first red)P(\text{first red})?

  • 5. Tree: two stages 1/21/2 then 1/31/3: path probability?

  • 6. Total probability: P(B)=P(B∣A)P(A)+P(B∣Ac)P(Ac)P(B)=P(B|A)P(A)+P(B|A^c)P(A^c) with all 1/21/2: P(B)P(B)?

  • 7. Median of {1,2,3,4,5}\{1,2,3,4,5\}?

  • 8. Arrangements of ABCABC: count?

  • 9. Independent events both 0.30.3: P(neither)P(\text{neither})?

Probability Practice Problems — Standard