Question

Question
a. A radioactive source is emitting, on the average, one particle per minute. If counting continues for several hundred minutes, during which time the particles are emitted randomly, in what proportion of these minutes is to be expected that
i. There will be 2 or more particles emitted? (3 Marks)
ii. There will be 2 or more particles emitted in 2 minutes? (3 Marks)

Answer 1

Let $X=$ the number of particles emitted: $X\sim Po(\lambda t)$

$$P(X=x)=\dfrac{e^{-\lambda t}(\lambda t) ^x}{x!}, =0, 1, 2, ...$$

i. $\lambda t=1$

$$P(X\geq2)=1-P(X=0)-P(X=1)=$$ $$=1-\dfrac{e^{-1}\cdot1^0}{0!}-\dfrac{e^{-1}\cdot1 ^1}{1!}=1-2e^{-1}$$

i. $\lambda t=1(2)=2$

$$P(X\geq2)=1-P(X=0)-P(X=1)=$$ $$=1-\dfrac{e^{-2}\cdot2^0}{0!}-\dfrac{e^{-2}\cdot2 ^1}{1!}=1-3e^{-2}$$

$$\dfrac{1-2e^{-1}}{1-3e^{-2}}=\dfrac{e^2-2e}{e^2-3}\approx0.445$$

$$\dfrac{1-3e^{-2}}{1-2e^{-1}}=\dfrac{e^2-3}{e^2-2e}\approx2.248$$