On average, there are 3 accidents per month at one intersection. We need to find out what is the probability that in any month at this intersectiona) What if there are 5 accidents? b) What if there are less than 3 accidents? c) What if there are at least 2 accidents?

Question

On average, there are 3 accidents per month at one intersection. We need to find out what is the probability that in any month at this intersectiona) What if there are 5 accidents?  b) What if there are less than 3 accidents?  c) What if there are at least 2 accidents?

berggansS · Accepted Answer

Given: mean&amp;nbsp;(λ)=3let &quot;X&quot; be the accidents per month occur at a certain intersectionX−Poisson(λ=3)formula:&amp;nbsp;P(X=x)=&amp;nbsp;e−λ&amp;nbsp;⋅&amp;nbsp;λxx!a)b)c)

Jeffrey Jordon · Accepted Answer

We will assume that the number of accidents at the intersection follows a Poisson distribution with a mean (μ) of 3 per montha)P[X=5]=p(5)=e−μμxx!=e−3355!=0.10082≈0.10b)P[X&amp;lt;3]=F(2)=∑x=02e−μμxx!=∑x=02e−33xx!=0.4232c)P[X≥2]=1−P[X&amp;lt;2]=1−P[X≤1]=1−F[1]=1−(e−3300!+e−3311!)=0.800852≈0.8

Accepted Answer

a)&amp;nbsp;λ=3, t=1, e=2.71828, X=5=(e^(-3))*(3^(5))&amp;nbsp;÷ 5!=12.0982576134 ÷ 120=0.10081881344&amp;nbsp;&amp;nbsp;b)X=0,1,2 as X&amp;lt;3p(0;3)+p(1;3)+p(2;3)p(0;3) = (e^(-3))*(3^(0))&amp;nbsp;÷ 0! = 0.04978706836 / 1 =&amp;nbsp;0.04978706836p(1;3) = (e^(-3))*(3^(1))&amp;nbsp;÷ 1! = 0.1493612051&amp;nbsp;p(2;3) = (e^(-3))*(3^(2))&amp;nbsp;÷ 2! = 0.44808361531 / 2 = 0.22404180765&amp;nbsp;p(0;3) + p(1;3) + p(2;3) =&amp;nbsp;0.04978706836 +&amp;nbsp;0.1493612051 +&amp;nbsp;0.22404180765 = 0.42319008111&amp;nbsp;&amp;nbsp;c)X&amp;gt;=2[1- ( p(0;3) + p(1;3) + p(2;3) ) ]&amp;nbsp;As I already derived the values for p(0;3), p(1;3) &amp;amp; p(2;3)I can directly solve it :1 -&amp;nbsp;0.42319008111&amp;nbsp;= 0.57680991889

On average, 3 traffic accidents per month occur at a certain intersection. What is the probability that in any given month at this intersection (a) exactly 5 accidents will occur? (b) fewer than 3 accidents will occur? (c) at least 2 accidents will occur?

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