Answer to Question #189840 in Statistics and Probability for LE10

Question

On average, 3 students per month cry in happiness. What is the probability that in any given month,

a) exactly 5 students will cry in happiness

b) fewer than 3 students will cry in happiness

c) at least 2 students will cry in happiness

Answers (Decimal Format; 4 decimal places)

Answer 1

Given, Average students cry in a month, $\lambda=3$

Let X denote the number of students cry in happiness

$(i) P(X=5)=\dfrac{e^{-\lambda}.\lambda^5}{5!}=\dfrac{e^{-3}3^5}{5!}=0.1008$

$(ii) P(X<3)=P(X=0)+P(X=1)+P(X=2)$

$=\dfrac{e^{-\lambda}.\lambda^1}{1!}+\dfrac{e^{-\lambda}.\lambda^2}{!}+\dfrac{e^{-\lambda}.\lambda^2}{2!}\\[9pt]=\dfrac{e^{-3}3^0}{0!}+\dfrac{e^{-3}3^1}{1!}+\dfrac{e^{-3}.3^2}{2!}\\[9pt]=0.048978+0.14936+0.22404=0.4224$

$(iii) P(X\ge 2)=1-P(X<2)=1-[P(X=0)+P(X=1)]$

$=1-[\dfrac{e^{-\lambda}\lambda^0}{0!}+\dfrac{e^{-\lambda}\lambda^1}{1!}] \\[9pt] =1-[0.048978+0.14936]\\[9pt]=1-0.198338=0.8016$