Homework SourseThe dynamics of the cougar population of Jasper National Par
The dynamics of the cougar population of Jasper National Park is being analyzed. A number of digital cameras and an image recognition system are used to automatically record cougar appearances throughout the park. Motion sensors detect large animals and set off the cameras, and after images are taken, they are sent to a computer for analysis using image recognition algorithms. Since there is no human involvement, mistakes do occur. In particular, 7% of animals that are not cougars are identified by the image recognition system as cougars, and also 16% of cougars are identified by the system as other animals. It is also known that, on average, only 6% of all the animals photographed are cougars.
a) What is the probability that the animal on a randomly selected image is identified as a cougar?
b) What is the probability that if an image is identified as that of a cougar, the animal on the image
actually is a cougar?
Can you show all steps,including the definitions.
Solution
let A be the event that a randomly selected animal is cougar. so Ac is the event that a randomly selected animal is not a cougar.
B be the event that the image identifies as cougar. so Bc is the event that image identifies as not cougar.
so B|A=the event that the image identies as cougar when it is actually cougar.
B|Ac=the event that the image identies as cougar when it is actually not a cougar.
Bc|A=the event that the image identies as not cougar when it is actually cougar.
Bc|Ac=the event that the image identies as not cougar when it is actually not cougar.
given that
P[B|Ac]=0.07 P[Bc|A]=0.16 P[A]=0.06
a)P[a randomly selected animal\'s image is identified as cougar]=P[B]=P[B|A]*P[A]+P[B|Ac]*P[Ac]=[1-P[Bc|A]]*P[A]+P[B|Ac]*[1-P[A]]=[1-0.16]*0.06+0.07*[1-0.06]=0.1162 [answer]
b) P[if image identified as cougar the animal is actually cougar]=P[A|B]
now by Bayes\' theorem
P[A|B]=P[B|A]*P[A]/P[B]=[1-P[Bc|A]*P[A]/P[B]=(1-0.16)*0.06/0.1162=0.4337 [answer]
