find the probability

Question

35 Blue (B); 20 Large (L), 15 Small (S)

30 Yellow (Y); 5L, 25S

20 Green (G); 15L, 5S

15 Red (R); 10L, 5S

 

 p(the frequency of Y in a sample of 4 selected with replacement = .25)

Answer 1

 

As we're going to pick up a ball, for example, with replacement,

the probability of picking up each different color each time will be

 

     prob(B) = 0.35,

     prob(Y) = 0.30,

     prob(G) = 0.20, and

     prob(R) = 0.15,

 

and the size of balls does not really matter in this case.

 

And, the probability of picking up a non-yellow is

 

     prob(non-Y) = prob (B) + prob(G) + prob(R)

                        =  0.35 + 0.20 + 0.15

                        = 0.70

 

So, the probability that the frequency of Y in a sample of 4 selected with replacement

equals 0.25 will be given by

 

     prob(the frequency of Y in a sample of 4 selected with replacement)

                        = ₄C₁ · prob(Y) · (prob(non-Y))³

                        = 4 · (0.30) · (0.70)³

                        = 0.4116

 

where ₄C₁ is the combinatorial function.

 

So, the answer is 0.4116.