The random variable X follows a Poisson distribution with mean $\lambda$.

(a)     Find $\lambda$ if ${\text{P}}(X = 0) + {\text{P}}(X = 1) = 0.123$.

(b)     With this value of $\lambda$, find ${\text{P}}(0 < X < 9)$.

(a)     required to solve ${{\text{e}}^{ - \lambda }} + \lambda {{\text{e}}^{ - \lambda }} = 0.123$     M1A1

solving to obtain $\lambda = 3.63$     A2     N2

Note: Award A2 if an additional negative solution is seen but A0 if only a negative solution is seen.

(b)     ${\text{P}}(0 < X < 9)$

$= {\text{P}}(X \leqslant 8) - {\text{P}}(X = 0)$ (or equivalent)     (M1)

$= 0.961$     A1

[6 marks]

Part (a) - Well done by most, although there were some answers that ignored the requirement of mathematical notation.

Part (b) - Not successfully answered by many. The main problem was not correctly interpreting the inequalities in the probability.