Two ponds, \(A\) and \(B\), each contain a large number of fish. It is known that \(2.4 \%\) of fish in pond \(A\) are carp and \(1.8 \%\) of fish in pond \(B\) are carp. Random samples of 50 fish from pond \(A\) and 60 fish from pond \(B\) are selected.
Use appropriate Poisson approximations to find the following probabilities.
The samples contain at least 2 carp from pond \(A\) and at least 2 carp from pond \(B\).
The samples contain at least 4 carp altogether.
The random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. It is given that
\(\mathrm { P } ( X = 0 ) = [ \mathrm { P } ( Y = 0 ) ] ^ { 2 }\),
\(\mathrm { P } ( X = 2 ) = k [ \mathrm { P } ( Y = 1 ) ] ^ { 2 }\), where \(k\) is a non-zero constant.
Find the value of \(k\).
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