Question

(Similar to Ghahramani Section 3.4, Problems 6, 8, and 14) Three black boxes are labeled with Roman numerals I, II and III.

• Box I contains four red chips and three blue chips.

• Box II contains two red chips and five blue chips.

• Box III contains seven red chips and no blue chips. Solve each of the following problems.

(a) Suppose a box is selected at random and three chips are drawn at random from the box. If all three chips are red, what is the probability they were drawn from Box I?

(b) Suppose one chip is selected at random from each box. If two of the three chips drawn are red chips, what is the probability that the chip drawn from Box II was red?

(c) Suppose three chips are randomly selected from Box I and placed in Box III. If a chip subsequently drawn randomly from Box III is blue, what is the probability that all three chips moved from Box I to Box III were blue.

Answer 1

a)

P(all 3 are red)=P(first box and all 3 are red)+P(2nd box and all 3 are red)+P(third box and all 3 are red)

=(1/3)*(4/7)*(3/6)*(2/5)+(1/3)*0+(1/3)*1 =13/35

hence P( Box I|all are red)=P(first box and all 3 are red)/P(all 3 are red)

=(1/3)*(4/7)*(3/6)*(2/5)/(13/35)=4/39

b)

P(2 are red out of 3)=P(red frm box 1 and 3 and blue from box 2)+P(red frm box 2 and 3 and blue from box 1)=(4/7)*1*(5/7)+(2/7)*1*(3/7)=26/49

P( Box II was red|2 are red out of 3)

=P(red frm box 2 and 3 and blue from box 1)/P(2 are red out of 3)

=(2/7)*1*(3/7)/(26/49)=6/26=3/13

c)

P(blue fom box III)=P(2 red and 1 blue chips from box I to III and blue from III)+P(1 red 2  blue chips from box I to III and blue from III)+P(0 red 3  blue chips from box I to III and blue from III)

=3*(4/7)*(3/6)*(2/5)*(1/10)+3*(3/7)*(2/6)*(4/5)*(1/10)*(2/10)+(3/7)*(2/6)*(1/5)*(3/10)=0.0497

probability that all three chips moved from Box I to Box III were blue given blue from box III

=P(0 red 3  blue chips from box I to III and blue from III)/P(blue fom box III)

=(3/7)*(2/6)*(1/5)*(3/10)/0.0497=5/29