Question

Two litters of a particular rodent species have been born; first with two brownhaired and one gray-haired (litter 1) and the other with three brown-haired and two gray-haired (litter 2). We select a litter at random and then select an offspring at random from the selected litter. (i) What is the probability that the animal chosen is brown-haired

Answer 1

The probability that the animal chosen is brown-haired is 0.6333.

Explanation:

Denote the events as follows:

A : a brown-haired rodent

B : Litter 1

The information provided is:

`P (A|B) =(2)/(3)\\\\P(A|B^(c))=(3)/(5)`

The probability of selecting any of the two litters is equal, i.e.

`P(B)=P(B^(c))=(1)/(2)`

According to the law of total probability:

`P(X)=P(X|Y_(1))P(Y_(1))+P(X|Y_(2))P(Y_(2))+...+P(X|Y_(n))P(Y_(n))`

Compute the total probability of event A as follows:

`P(A)=P(A|B)P(B)+P(A|B^(c))P(B^(c))`

`=[(2)/(3)*(1)/(2)]+[(3)/(5)*(1)/(2)]\\\\=(1)/(3)+(3)/(10)\\\\=(10+9)/(30)\\\\=(19)/(30)\\\\=0.6333`

Thus, the probability that the animal chosen is brown-haired is 0.6333.