Probability Distributions and Binomial Distribution
Probability Distributions and Binomial Distribution tests discrete random variables, probability mass functions, expectation and variance, and the conditions and calculations of the binomial model. The exam often combines a distribution table with a binomial word problem.
Core rules
- For a discrete random variable X, P(X=x_i)=p_i, each p_i≥0 and Σp_i=1.
- Mean or expectation E(X)=Σx_i p_i. Variance Var(X)=Σx_i^2 p_i - [E(X)]^2.
- X follows B(n,p) when there are n independent identical trials, each with constant success probability p and two outcomes.
- Binomial probability: P(X=r)=nCr p^r (1-p)^(n-r), r=0,1,...,n.
- For X~B(n,p), E(X)=np and Var(X)=np(1-p); standard deviation is √(np(1-p)).
- At least one means 1-P(X=0); at most r means P(X=0)+...+P(X=r).
Worked example
If X~B(5,0.4), then P(X=2)=5C2(0.4)^2(0.6)^3=10×0.16×0.216=0.3456. Also E(X)=5×0.4=2 and Var(X)=5×0.4×0.6=1.2. The complement is useful: P(X≥1)=1-(0.6)^5.
Mark-losing traps
- Using nCr when order matters, or omitting it when choosing r successes.
- Mixing up p and 1-p in the failure factor.
- Calculating exactly two when the question says at least two.
- Using binomial variance np instead of np(1-p).
- Assuming trials are binomial when they are not independent or p changes.