Each week, Eric is supposed to prepare 4 exercises for the following week’s home

Question

Each week, Eric is supposed to prepare 4 exercises for the following week’s homework assignment. There is also a 53% chance that a visiting scholar arrives each week and engages Eric with numerous research questions. During the weeks when the scholar is visiting, the number of problems Eric actually writes is Poisson distributed with a mean of 3 exercises. The number of problems Eric actually creates in a week with no visitors follows a Poisson distribution with mean 6.

a. What is the probability that Eric manages to create enough exercises for the following week's homework during a week with no visitors? Round your answer to 4 decimal places.

b. If Eric fails to write 4 exercises one week, what is the probability that he received a visiting scholar that week? Round your answer to 4 decimal places.

c. In the last week of the semester, Eric decides to assign every exercise he writes. Given that Eric writes five exercises, what is the probability that Eric did not receive a visitor in the last week of the semester? Round your answer to 4 decimal places.

Explanation / Answer

a) here parameter =6

probability that Eric manages to create enough exercises for the following week's homework =P(X>=4)

=1-P(X<=3) =1-x=03 e- x/x! =1-0.1512 =0.8488

b)

)probabilty that eric fails to write 4 problems =P(  visiting scholar arrives and fails to write + visiting scholar does not arrives and fails to write) = 0.53*x=03 e-11 x/x! +0.47*x=03 e- x/x!    where 1 =3

         =0.53*0.6472 +0.6*0.1512 =0.4141

therfore probability that he received a visiting scholar that week given fails to write 4 exercises

==P(  visiting scholar arrives and fails to write)/P( falils to write) =0.53*0.6472/0.4141 =0.8284

c)probability he write 5 excercises =P(  visiting scholar arrives and writes 5 excercise + visiting scholar does not arrives and writes 5 excercise)=P(X=5)= 0.53* e-11 x/x! +0.47* e- x/x! =0.53*0.1008+0.47*0.1606

        =0.1289

therefore  probability that Eric did not receive a visitor given writes five exercises =0.53*0.1008/0.1289=0.4145

please revert for any clarification required

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