Question

A soldier who hits a target with probability 2/3 fires shots until he hits the target or 5 shots have been fired. Let X denote the number of shots fired. Find the
i) probability distribution of X
ii) mean of X

Answer 1

The number of shots fired

$$X={1,2,3,4,5}$$

Probability of hitting the target

$$P = \frac{2}{3}, \ (1-P) = \frac{1}{3}$$

The probability of hitting the target and stopping given X number of shots

$$P (X=x) = P(1-P)^{x-1}$$

i)

$$P (X=1) = \frac{2}{3}$$ $$P(X=2) = \frac{2}{3} * \frac{1}{3} = \frac{2}{9}$$ $$P(X=3) = \frac{2}{3} * (\frac{1}{3}) ^2 = \frac{2}{27}$$

$$P(X=4) = \frac{2}{3} * (\frac{1}{3}) ^3 = \frac{2}{81}$$

$$P(X=5) = \frac{2}{3} * (\frac{1}{3}) ^4 + (\frac{1}{3})^5 = \frac{3}{243}$$

ii)

$$E(X) = \sum x p(x)$$

$$1 *\frac{2}{3} + 2 * \frac{2}{9} + 3*\frac{2}{27}+4*\frac{2}{81}+ 5*\frac{3}{243}$$

$$E(X)=1.4938$$