In addition to zebra, Professor Getz\'s group studied springbok (a type of antel

Question

In addition to zebra, Professor Getz's group studied springbok (a type of antelope) in Etosha National Park. In a study paralleling Example 2 of this section, they found that the probability of a springbok surviving predation and disease to the end of the eighth week of a typical year is given by, respectively, 94.9% and 98.8%. Suppose they also determined that at the end of this eight weeks the probability of dying from predation and disease is decreasing respectively by 0.6% and 0.25% per week. Assume the events of dying from predation or disease are independent.

a. Find the probability of surviving to the end of week 8.

b. Find the rate of change of probability of surviving at the end of week 8.

c. Estimate the probability of surviving to the end of week 9.

Explanation / Answer

let A be the event that a springbok surviving predation at the end of 8th week

B be the event that a springbok surviving disease at the end of 8th week

hence Ac be the event that a springbok dying from predation at the end of 8th week

Bc be the event that a springbok dying from disease at the end of 8th week

Assume the events of dying from predation or disease are independent.

that is Ac and Bc are independent. hence A and B are also independent

the probability of a springbok surviving predation and disease to the end of the eighth week of a typical year is given by, respectively, 94.9% and 98.8%

hence P[A]=0.949 and P[B]=0.988

a) so probability of surviving to the end of week 8 is

P[A or B]=P[A]+P[B]-P[A and B]

              =P[A]+P[B]-P[A]*P[B] [since they are independent]

              =0.999388 [answer]

b) hence P[Ac]=1-P[A]=1-0.949=0.051

P[Bc]=1-P[B]=1-0.988=0.012

they also determined that at the end of this eight weeks the probability of dying from predation and disease is decreasing respectively by 0.6% and 0.25% per week

hence the new probability of dying from predation is P[Acn]=0.051-0.051*0.6/100=0.050694

hence the new probability of dying from disease is P[Bcn]=0.012-0.012*0.25/100=0.01197

so P[An]=1-0.050694=0.949306 P[Bn]=1-0.01197=0.98803

hence the new probability of surviving is

P[An or Bn]=P[An]+P[Bn]-P[An and Bn]=0.949306+0.98803-0.949306*0.98803=0.999393

hence the rate of change is (0.999393-0.999388)*100/0.999388=0.0005 %   [answer]

c) the probability of surviving at the end of week 9 is

P[An or Bn]=P[An]+P[Bn]-P[An and Bn]=0.949306+0.98803-0.949306*0.98803=0.999393 [answer]

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.