Probability and Statistics
Probability and statistics
Know classical probability, the addition and multiplication rules, conditional probability, independence, and the complement rule. For a random variable, compute the distribution, expectation, and variance; for binomial settings, use C(n,k)p^k(1−p)^{n−k}. Understand sampling, the difference between mean and median, and how to read frequency tables and simple regression output. Conditional probability is P(A|B)=P(A∩B)/P(B), valid when P(B)>0.
Traps: treating draws without replacement as independent; using “at least one” directly instead of its complement; confusing mutually exclusive events with independent events; and forgetting to restrict the sample space in a conditional probability. Also check whether a probability mass function sums to 1 and whether expectation is weighted by probability, not by outcome count.
Worked example: a bag has 3 red and 2 blue balls. Two are drawn without replacement. The probability the second is red is P(R₂)=P(R₁)P(R₂|R₁)+P(B₁)P(R₂|B₁)=(3/5)(2/4)+(2/5)(3/4)=3/5, matching symmetry. For “at least one red,” use 1−P(no red)=1−(2/5)(1/4)=9/10. Rule of thumb: name the events, write the conditioning event in the denominator, and use complements for “at least.”