If on an average $1$ ship out of $10$ does not arrive safely at a port, find the mean and standard deviation of ships returning safely out of a total of 500 ships.
Probability of a ship not arriving safely at a port $= q = \dfrac{1}{10}$
$\therefore \;$ Probability of a ship arriving safely at a port $= p = 1 - q = 1 - \dfrac{1}{10} = \dfrac{9}{10}$
Total number of ships $= n = 500$
Mean number of ships arriving safely at port $= n \times p$
$\therefore \;$ Mean number of ships arriving safely $= 500 \times \dfrac{9}{10} = 450$ ships
Standard deviation of number of ships arriving safely at port $= \sqrt{n \times p \times q}$
$\therefore \;$ Required standard deviation $= \sqrt{500 \times \dfrac{9}{10} \times \dfrac{1}{10}} = 3 \sqrt{5}$