Probability (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · Gaokao

Probability: core idea

Conditional tables.

Bayes: disease prior $0.01$ , test + if sick $0.99$ , false + $0.05$ . Approx posterior? Enter nearest percent.

Hint: Classic Bayes ~16.5%. Answer: 17.

1. A binomial random variable has $n=100$ trials and success probability $p=0.5$ . What is its mean?
2. Covariance independent?
3. Expected return: win $10$ with $p=0.4$ else $0$ : expectation?
4. Assuming 365 equally likely birthdays, is the probability that at least two of 23 people share a birthday greater than $1/2$ ? Enter 1 for yes or 0 for no.
5. Geometric $p=1/3$ : expected trials to first success?
6. Frequency: 45 heads in 100: estimate $p$ ?
7. Conditional expectation: $E[X|X>0]$ for fair die — average of $1..6$ ?
8. Model: Poisson $\lambda=1$ : $P(0)$ ?
9. Repeated: $P(\text{at least one 6 in 2 dice})$ ?