Question

4. Workers in a factory incur accidents at a mean rate of 2 accidents per week. ICA a) What is the probability that there are (i) at most 2 accidents in a given week? (ii) at least 4 accidents in a given week? b) Out of 5 weeks, what is the probability that at most 2 of the weeks will have at least 4 accidents? [15 marks]

Answer 1

a)

here for poisson distribution expected accidents in a week=2= $\lambda$

i)P(at most 2 accidents)=P(X<=2)=P(X=0)+P(X=1)+P(X=2)=e^-22⁰/0!+e^-22¹/1!+e^-22²/2!=0.6767

ii)P(at least 4 accidents)=P(X>=4)=1-P(X<=3)=1-(P(X=0)+P(X=1)+P(X=2)+P(X=3))

=1-(e^-22⁰/0!+e^-22¹/1!+e^-22²/2!+e^-22³/3!) =1-0.8571=0.1429

b)

from binomial distribution:

P(at most 2 weeks with at least 4 accidents)=P(Y<=2)=P(Y=0)+P(Y=1)+P(Y=2)

=⁵C₀(0.1429)⁰(0.8571)⁵+⁵C₁(0.1429)¹(0.8571)⁴+⁵C₂(0.1429)²(0.8571)³=0.9767