1 Three balls are randomly drawn without replacement from a

1. Three balls are randomly drawn (without replacement) from a bowl containing 4 red balls and 6 yellow balls. Let X= the number of yellow balls drawn.

a) How is X distributed? (Give the name of the distribution.)

b) What is the probability that exactly 2 of the three balls are yellow?

c) What is the probability that less than 2 of the three balls are yellow?

d) What is the probability that none of the three balls are red?

Please do a step by step through each part

Solution

A) Hypergeometric distribution

B) we have to find the probability that exactly 2 of the three balls are yellow

   let Y denotes Yellow ball and R denotes Red ball

then our favourable cases RYY,YRY,YYR

then P(X=2)=P(R)P(Y|R)P(Y|Y,R)+P(Y)P(R|Y)P(Y|Y,R)+P(Y)P(Y|Y)P(R|Y,Y)

                 =(4/10)*(6/9)*(5/8)+(6/10)*(4/9)*(5/8)+(6/10)*(5/9)*(4/8)

                    =(1/6)+(1/6)+(1/6)=1/2 =0.5answer

C) we have to find probability that less than 2 of the three balls are yellow

means we have to find P(X<2)

means X =0 or X=1

then favourable case

RRR,YRR,RYR,RRY

P(X<2)=P(R)P(R|R)P(R|R,R)+P(Y)P(R|Y)P(R|R,Y)+P(R)P(Y|R)P(R|,Y)+P(R)P(R|R)P(Y|R,R)

        =(4/10)*(3/9)*(2/8)+(6/10)*(4/9)*(3/8)+(4/10)*(6/9)*(3/8)+(4/10)*(3/9)*(6/8)

       =(1/30)+(1/10)+(1/10)+(1/10)

           =10/30=1/3=0.33 answer

D)we have to find the probability that none of the three balls are red

mean we have to find P(X=3)

YYY

then

P(X=3)=P(Y)P(Y|Y)P(Y|Y,Y)

          =(6/10)*(5/9)*(4/8)=1/6=0.166 answer