Question

In an urn, there are 10 blue balls and 10 red balls. Balls are randomly selected from the urn over and over again. Find the probability that the third blue ball is chosen on the fifth selection under the different assumptions: a) Balls are NOT returned to the urn after selection. b) Balls are returned to the urn after being selected.

Answer 1

Prob. of selecting Blue ball in the est trial a selecting I Blue sall out of to Blue balls (favorable cases) y selecting I ball out of 20 total balls ( Possible cases) 10 and trial 1o ci 20 C, 20 2 gci 19 CI Next, prob. of selecting a Blue ball in the Į blue ball decreased and total also decreased by 1. =9 19 Prob. of Red ball Curite this is the 3rd trial) selecting I Red ball out of 10 Red balls selecting I ball out of total 18 balls C Two missing balls were blue) a selecting oci 1861 Wor 18 Prob. of selecting 9c itci 9 17 Red ball in the 4th trial I red ball decreased and total balls decrease by 1 with each trial. Blue ball in the 5th trial. Prob. of gelecting a 8(1 NI- 16. 45 166, so, total poob. Х x x 9 x 17 2 2 1292

S cases can similarly, the rest also happen. Calculate their probabilities using the concept. 1st, 3rd and 5th x 8 x 2 loc zoci 10C TICI X Х (9c, x T8(1) 9c, x TICI 18 420 x Lol X 9 118 It 162 45 1 x 4 Г 19 x 17 1292 1°C 1st, 4th and sth (8L1 ali a 1°C18 1911 gci T8(1 20(1 Ali 16(1 45 1292 1°C X oli toCy 94 1811 aux T6L 2nd, 3rd and 5th = ** 19Ci Ali 2001 - 45 1292 1061 x lisci loci 9C1 X IBC 19C1 18C1 2nd, 4th and 5th = 2001 iPa TECH 45 1292 3rd, 4th and 8th loc1 x 2001 9C1 19C1 1961X 18C 9C1 in 1661 + 45 45 1292 Total prob. = Either of these 6 cases 45 45 45 + 45 + /292 1292 1992 1292 1292 270 1292 1292 45 + + 1292 = 0.209 (Ans.) 6 x 45 (Note : Blue ball selections have been encircled)

Balls are returned » Total balls in the win remain constant with the selections. selection. Again, consider the case of 1st, 2nd and 5th Ist (Blue Ball) prob. I (Blue Ball) poob. lo loc 2ol1 = 2 2nd loci 2001 un 20 2 (Now note, no. of balls didn't change as the ball returned to the urn) Similarly, all other 3 probs. in this case will be loc, 2001 ed af Reda Red ball each. Σ - = case = *** Blute ball / I ; Total prob. for this 32 Blure Blue all Combining ball the cases bay Potal prob. Either of the 6 cases (where prob. of each case case is 1/2) I tl tl tl f eff 32 32 32 = 6XL 3 0.1895 (Ans.) 16 32 32 32 32 Kindly put a Thumbs up!