Probability and Statistics
Probability and Statistics tests classical and conditional probability, independent events, Bayes theorem, random variables, binomial distribution, and measures of central tendency and dispersion.
Core rules
- P(A|B)=P(A∩B)/P(B), with P(B)>0. Events are independent when P(A∩B)=P(A)P(B).
- For a binomial variable with n trials and success probability p, P(X=r)=nCr p^r(1-p)^(n-r), mean=np and variance=np(1-p).
- Mean of data is sum of values divided by number of observations; variance is the mean of squared deviations from the mean; standard deviation is its square root.
- For grouped data, use class midpoints and frequencies; check whether the question asks for population or sample variance.
Worked example
A fair coin is tossed 4 times. Find P(exactly 2 heads). Here n=4, p=1/2, r=2. P=4C2(1/2)^2(1/2)^2=6/16=3/8. The mean number of heads is np=2 and variance is np(1-p)=1.
Traps
- Mutually exclusive events are not independent unless one has probability zero.
- In conditional probability, the denominator is P(B), not the total sample space count.
- Do not forget the binomial coefficient nCr.
- Variance uses squared deviations; mean absolute deviation is a different measure.
- In Bayes theorem, keep the prior and likelihood terms in the correct numerator and use total probability in the denominator.