Question

An ogre is very hungry but cannot decide how many chickens he wants to eat. So, he rolls a standard 6-sided die. Let Y be the number that he rolls. Then he goes to his chicken coop and randomly chooses Y chickens to eat. Suppose that his coop has 10 Rhode Island Reds and 4 Bantams.

(a) What is the probability that all of the Bantams get eaten? (b) What is the expected number of Bantams consumed?

Answer 1

Y denotes the number of chickens ogre chooses

Let X denote the number of Bantams chosen

Y is actually the number on the rolled die

P(Y = k) = 1/6, k = 1,2,3,4,5,6

Coop has 10 Rhode Island Reds and 4 Bantams

==> X can take values 0,1,2,3,4

The probability that x bantams are chosen when y chickens are to be eaten

P(X,Y) = P(X|Y).P(Y) Bayes rule

a)

   since minimum 4 chickens must be chosen for all bantams to be eaten

Therefore the probability that all of the Bantams get eaten is 0.0035

b)

​​​​​​​

​​​​​​​

X	0	1	2	3	4
P(X)	0.3240	0.4091	0.2133	0.0501	0.0035

Therefore the expected number of Bantams consumed is 1