
Thumbtack Problem
When a thumbtack is flipped, the probability that it lands up is 0.6. The thumbtack is flipped five times. a) Find the probability that the thumbtack lands up exactly twice in the five flips. b) Find the probability that the thumbtack lands up at least twice in the five flips. c) Let X be the number of times the thumbtack lands up in the five flips. Find the mean \mu (x), and the variance Var(X).

(a) If $\displaystyle X$ is the number of times the thumbtack lands up, $\displaystyle X$ can be said to follow a $\displaystyle Bin(5,0.6)$ distribution. Calculate $\displaystyle P(X=2)$ from this.
(b) Calculate $\displaystyle P(X\geq 2)$
(c) If $\displaystyle X\sim Bin(n,p)$ , $\displaystyle E(X)=np$ , $\displaystyle Var(X)=np(1p)$.....