Random Signals for Engineers using MATLAB and Mathcad Copyright 1999 Springer Verlag NY Example 1.9 In this example we will calculate the probability of drawing 4 red balls from a bin that contains 4 red balls and 46 white balls. We will draw the balls assuming that the individual draws are independent and then recalculate the probabilities when the draws are not. Replacing the ball in the bin every time a draw is made can simulate the independent case. SOLUTION: Independent Case -The probability of drawing 4 red balls is just the product of 4 draws where the P[a red ball is drawn] = Pr . Pr=4/50; Since there are 4 red balls available for each draw from the 50 total in the bin, the probability of drawing a red ball is P[A] and the probability of drawing 4 red balls on 4 tries is PA1 A2 A3 A4 PA1 PA2 PA3 PA4 where Ai is the event that represents drawing a red ball. If the ball is replaced for the next draw, when a new ball is drawn the conditions are the same as the previous draw. We have an independent draw since the present draw does not depend on the previous one. The P[four red balls in 4 draws] is Pr^4 ans = 4.0960e-005 Dependent Case - If we do not replace the balls we will use the laws of conditioned probabilities to compute the P[four red balls in 4 draws]. P A A P A A P A A A P A A P A P A1 A2 A3 A4 P A2 A3 A4 A1 P A1 P A3 A4 A1 A2 P A2 A1 P A1 1 A2 A3 4 4 A1 A2 3 3 1 2 2 1 1 This equation states that the second draw is conditioned on the first and the third is conditioned on the first two and the forth is conditioned on the preceeding three. P[A1] is Pr ; P[A2|A1] is Pr1; P[A3|A1 A2] = Pr2 and P[A4|A1 A2 A3] = Pr3. The probabilities are computed by ratio of ways we can select a red ball over the number of ball in the bin when each red ball is drawn. Pr1=3/49; Pr2=2/48; Pr3=1/47; In all the cases we reduce the red ball count by 1 and the number in the bin by 1 as we draw red balls out of the bin. The probability of 4 draws without replacement is Pr*Pr1*Pr2*Pr3 ans = 4.3422e-006 The values are not the same. For the conditioned case, the probability of drawing a red ball decreases as we draw red balls and thus the overall probability of drawing 4 red balls conditioned draw is less than the independent-draw case.