Question

please use the appropriate probability notation and formulas when presenting and label importsnt variables. will upvote!

4. Blindfolded Musical Chairs We are playing a game of blindfolded musical chairs with 20 blindfolded people and 40 chairs. When the music stops each person picks a chair uniformly at random and sits on it. 1. What is the probability that some chair has at least two people sitting on it? 2. What is the probability that some chair has at least three people sitting on it?

Answer 1

Sol. I Let Total number of blindfolded people = 20 Total number of chairs = 40 the all chairs are numbered as number - 1,2,...,40 when the music stops each person picks a chair uniformly at sandom & sits on it o denete a dove as xq which is defined as - X. : Number on the chair choosen by . . the ith person ; i = 1, 2, ...,20 - Xe can take 1,2,---,40 Since each & person picks a choir 1 uniformly at senden i x; follows iid discrete wuiform disterbution ovet (1,40] A Pl. Some chair has at least two people ] None of the choir contains more than one person ]

= 1-P ( x + x2 + xz #... #Xzo) i = 1 12 =1 iz=1 - isti, iztiz ti, izo=1 isotingt. Il - 1- 40x39x38x---X21 (40)° 3 8 This is the ② define required probability. X = 1 : 1 Vizioa Hi=1(1)20 4081 istith position 0 / 40x 1 40X1 S = x2+xz +---+ Xizo Ş= | SA 40x1 | Sz Stolfox1 where S; denote number of person on ith chair ; v j = 1, 2, ....,to shifting On j= 1,2,---- 4-0

. Ş follows a multinomial dist" with number of varsiables (n)=20 number of Components in each variable (!)=40 € & probability of Success Ly, ,L40 each - is equal to ...no P[ Some chair has at least three peldens] = = 1-P [None of the chair Contains - more than two person] = 1-P ($452; 522; ---, S4052] each person picks a chair at sondom in each si are independent with each other P[ Some cheis has at least 3 person] = 1- P(52). P(5,52)..-- P (5,52). = 1-(3) *