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Question

Six person try to swim across a wide river. It is known that on an average, only three persons out of the ten are successful in crossing the river. what is the probability that at most four of the six persons will cross river safely?


A
0.990523
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B
0.890523
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C
0.980523
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D
None of these
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Solution

The correct option is A $$0.990523$$
Let success be defined as crossing the river.
$$\Rightarrow $$ probability of success $$ \displaystyle \left( p \right) =\frac { 3 }{ 10 } $$ and that of failure $$ \displaystyle \left( q \right) =\frac { 7 }{ 10 }  $$
Since number of trials $$(n)=6$$
$$\therefore P$$ (at most $$4$$ success in $$6$$ trials) $$=1-^{ 6 }{ { C }_{ 5 }{ P }^{ 2 }q- }^{ 6 }{ { C }_{ 6 } }{ P }^{ 6 } $$
$$ \displaystyle =1-6{ \left( \frac { 3 }{ 10 }  \right)  }^{ 5 }\left( \frac { 7 }{ 10 }  \right) -{ \left( \frac { 3 }{ 10 }  \right)  }^{ 6 }=1-{ \left( \frac { 3 }{ 10 }  \right)  }^{ 6 }\left( 13 \right) =0.9905 $$

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