Question Hypothetical A particular theory in genetics predi

Question : (Hypothetical) A particular theory in genetics predicts that a certain genetic trait, call it the ‘X’ trait, occurs in a particular species at a rate of 1 in every 5. A random sample of n = 4 animals of this species is selected and the number X with this trait (in the sample) is recorded.

a)Find the probability that exactly one will have the trait Hint: you could use the notation (X,0,0,O) Ans:________________

b)Find the probability that exactly 2 will have the X-trait {as in (X, X, 0, O) or (XOXO) Ans:________________

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where     
k = number of successes in trials        
n = is the number of independent trials        
p = probability of success on each trial

a)
P( X = 1 ) = ( 4 1 ) * ( 0.2^1) * ( 1 - 0.2 )^3
= 0.4096
b)
P( X = 2 ) = ( 4 2 ) * ( 0.2^2) * ( 1 - 0.2 )^2
= 0.1536