Standard +0.8 This question requires understanding that independent Poisson distributions sum to another Poisson, then correctly calculating the combined parameter: 2 minutes × (3×1.5 + 2×2) = 17, followed by evaluating P(X=27) from Po(17). While conceptually straightforward for students who know the sum property, it involves multiple layers of calculation and careful parameter arithmetic, making it moderately challenging but still within standard S3 scope.
1 The numbers of \(\alpha\)-particles emitted per minute from two types of source, \(A\) and \(B\), have the distributions \(\operatorname { Po } ( 1.5 )\) and \(\operatorname { Po } ( 2 )\) respectively. The total number of \(\alpha\)-particles emitted over a period of 2 minutes from three sources of type \(A\) and two sources of type \(B\), all of which are independent, is denoted by \(X\). Calculate \(\mathrm { P } ( X = 27 )\).
1 The numbers of $\alpha$-particles emitted per minute from two types of source, $A$ and $B$, have the distributions $\operatorname { Po } ( 1.5 )$ and $\operatorname { Po } ( 2 )$ respectively. The total number of $\alpha$-particles emitted over a period of 2 minutes from three sources of type $A$ and two sources of type $B$, all of which are independent, is denoted by $X$. Calculate $\mathrm { P } ( X = 27 )$.
\hfill \mbox{\textit{OCR S3 2006 Q1 [4]}}