A fisherman notices that in any hour of fishing, he is equally likely to catch exactly two fish, as he is to catch less than two fish. Assuming the number of fish caught can be modelled by a Poisson distribution, calculate the expected value of the number of fish caught when he spends four hours fishing.

\(X \sim {\text{Po}}(m)\)

\({\text{P}}(X = 2) = {\text{P}}(X < 2)\)     (M1) 

\(\frac{1}{2}{m^2}{{\text{e}}^{ - m}} = {{\text{e}}^{ - m}}(1 + m)\)     (A1)(A1)

\(m = 2.73     {\text{    }}\left( {1 + \sqrt 3 } \right)\)     A1

in four hours the expected value is 10.9\(\,\,\,\,\left( {4 + 4\sqrt 3 } \right)\)     A1

Note: Value of m does not need to be rounded.

 [5 marks]

Many candidates did not attempt this question and many others did not go beyond setting the equation up. Among the ones who attempted to solve the equation, once again, very few candidates took real advantage of GDC use to obtain the correct answer.