Question
If the chance that a vessel arrives safely at a port is $$\frac{{9}}{{10}}$$ then what is the chance that out of 5 vessels expected at least 4 will arrive safely?
Answer: Option A
The probability that exactly 4 vessels arrive safely is,
$${ = ^5}{C_4} \times {\left( {\frac{9}{{10}}} \right)^4}\left( {\frac{1}{{10}}} \right)$$
The probability that all 5 arrive safely is $${\left( {\frac{9}{{10}}} \right)^5}$$
The probability that at-least 4 vessels arrive safely,
$$ = {}^5{C_4} \times {\left( {\frac{9}{{10}}} \right)^4}\left( {\frac{1}{{10}}} \right) + $$ $${\left( {\frac{9}{{10}}} \right)^5}$$
$$ = \frac{{14 \times {9^4}}}{{{{10}^5}}}$$
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The probability that exactly 4 vessels arrive safely is,
$${ = ^5}{C_4} \times {\left( {\frac{9}{{10}}} \right)^4}\left( {\frac{1}{{10}}} \right)$$
The probability that all 5 arrive safely is $${\left( {\frac{9}{{10}}} \right)^5}$$
The probability that at-least 4 vessels arrive safely,
$$ = {}^5{C_4} \times {\left( {\frac{9}{{10}}} \right)^4}\left( {\frac{1}{{10}}} \right) + $$ $${\left( {\frac{9}{{10}}} \right)^5}$$
$$ = \frac{{14 \times {9^4}}}{{{{10}^5}}}$$
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