Question

A sheepdog trial requires the sheepdog to fetch sheep to the handler, drive sheep away from the handler, herd the sheep into a pen, and separate the sheep into two groups. A good sheepdog can fetch the sheep successfully with probability .9, drive sheep away successfully with probability .7, herd the sheep into a pen successfully with probability .84, and separate the sheep into groups successfully with probability .75. The ability of any sheepdog to perform any one of these tasks is independent from its ability to perform any other of them (or any combination of the others).

1. What is the probability that a sheepdog performs exactly three of these tasks successfully?

2. What is the probability that a sheepdog performs at least one of them successfully?

Answer 1

Assume Fetch sheep to the handler - A

Drive sheep away from the handler - B

Herd the sheep into a pen- C

Separate the sheep into two groups -D

So P(A)=0.9

P(B)=0.7

P(C)=0.84

P(D)=0.75

1. p( a sheepdog performs exactly three of these tasks successfully)=

=0.9*0.7*0.84*0.25+0.9*0.7*0.16*0.75+0.9*0.3*0.84*0.75+0.1*0.7*0.84*0.7

=0.1323+0.0756+0.1701+0.0441

=0.4221

P(Atleast one of them successful)=1-P(None of them are successful)

=

=1-0.1*0.3*0.16*0.25

=1-0.0012

=0.9988

Hope this will be helpful. Thanks and God bless You :)