SL 4.8—Binomial distribution

X~B(n,p) models the number of successes in n trials when each trial has two outcomes, trials are independent, n is fixed and the success probability p stays constant. P(X=r)=C(n,r)p^r(1−p)^(n−r).

The modelling decision comes before the calculator command. If trials affect one another, have more than two outcomes, or p changes, the binomial assumptions fail even if the question uses the word ‘success’.

If a sensor fails with probability 0.1 independently on each of 20 checks, the number of failures is B(20,0.1). ‘At least two’ means P(X≥2), not P(X=2), so the event wording controls the calculation.

A binomial distribution is not any count between 0 and n. State the trial, success definition, independence and constant-p assumption explicitly.

For $X\sim B(n,p)$, $E(X)=np$ and $Var(X)=np(1-p)$, so $SD(X)=\sqrt{np(1-p)}$. In the sensor example, $E(X)=20(0.1)=2$ failures and $Var(X)=20(0.1)(0.9)=1.8$. Use technology for exact, cumulative or interval probabilities and translate phrases such as 'at most' and 'more than' into inequalities before entering values.