Polya’s urn scheme: You select a ball at random from an urn containing 4 red bal
ID: 3329735 • Letter: P
Question
Polya’s urn scheme: You select a ball at random from an urn containing 4 red balls and 4 green balls. Then you replace that ball along with 2 more of the same color, and then you again select a ball at random. Are the events {first ball selected is red} and {second ball selected is red} independent? Justify your answer in two different ways with appropriate probability calculations.
6. Polya's urn scheme: You select a ball at random from an urn containing 4 red balls and 4 green balls. Then you replace that ball along with 2 more of the same color, and then you again select a ball at random. Are the events (first ball selected is red and {second ball selected is red} independent? Justify your answer in two different ways with appropriate probability calculationsExplanation / Answer
Total number of balls =8
red balls =4
green balls =4
1st draw
P(drawing red ball) =4/8 =0.5 P(drawing green ball) =4/8 =0.5
after first draw
total number of balls =10
Number of red balls =6 (when in first draw we got red ball)
=4 (when in first draw we got green ball)
Number of green balls =6 (when in first draw we got green ball)
=4 (when in first draw we got red ball)
Now
P(getting red ball in second draw) =P(getting red ball in 2nd draw | in first draw we got red ball)*P(1st draw red)
+P(getting red ball 2nd draw | in first draw we got green ball)*P(1st draw green)
=(6/10)*0.5 +(4/10)*0.5
=0.30+0.20=0.5
Now
P(red in 1st draw and in second draw)=P(getting red ball in 2nd draw | in first draw we got red ball)*P(1st draw red)
=(6/10) *0.5 =0.3
now
P(red in 1st draw)*P(red in second draw) =0.5*0.5 =0.25 < 0.3
since P(red in 1st draw)*P(red in second draw) not equal to P(red in 1st draw and in second draw) hence events {first ball selected is red } and {second ball selected is red} are not independent .