Question

In a box, there are 400 blue marbles and 100 red marbles. When 200 marbles are randomly chosen from the box, let the random variable X represent the number of chosen blue marbles. (a) [10 points) Find P(X > 102). [You may leave the answer in terms of "combinations” or “factorials”.) (b) [10 points) Find P(X = 160) using binomial approximation.

Answer 1

solution: - Given that :- Blue marble are 400 Red marble 100 . The Randomly chosen mondes from the box probability for selecting toure marsles is 400 - 0.80 400+100 probability of red marbles is 100 400 +L00 - 0-10 Tou yourdom variable x represent bember of chosen blue marbles. a) using using winomial approximation P (x = 102) Ż (2.20) 0.80 (1-0-80) (en x = 102) . (substitute x= 102) 200-a x= 102 20.987 N 1 Normal approximation the Expected number of blue toalls out of 200 is = 0.80 x 200 2160 . The varience of the Number of blere yally out of 200 = 0.80 x 0.20 x 200 32

P (x = 102) X - u 102 - 160 60) 6 √32 & P (2 - 10.253 0.987 & 1 b) binomial approximation, the probability 160 P(x = 160) - (200) 0.6010 (1-0.40)200-160 Pa = 160) 0.0704 Note: - using typer geomerie distribution, we have P (x = 1687 "z typgeom. Dist (160, 200, 400, 500,0)=0.09.07