Question

If on an average $9$ ships out of $10$ return safe to port, what is the chance that out of $5$ ships expected at least $3$ will arrive?

Answer 1

If on an average $9$ ships return from $10$ ships. Hence, probability of safe return is $\frac{9}{10}$.

Out of $5$ ships atleast $3$ to be returning with safety is:

a) Exactly 3 ships returning with safety:${}^5{C}_3\times {\left(\frac{9}{10}\right)}^3\times {\left(\frac{1}{10}\right)}^2=\frac{7290}{{10}^5}$

b) Exactly 4 ships returning with safety:${}^5{C}_4\times {\left(\frac{9}{10}\right)}^4\times {\left(\frac{1}{10}\right)}^1=\frac{32805}{{10}^5}$

c) Exactly 5 ships returning with safety:${}^5{C}_5\times {\left(\frac{9}{10}\right)}^5\times {\left(\frac{1}{10}\right)}^0=\frac{59049}{{10}^5}$

Hence, the total probability is $=\frac{7290}{{10}^5}+\frac{32805}{{10}^5}+\frac{59049}{{10}^5}=\frac{99144}{{10}^5}=0.99$