Question

3. Suppose we have a late night bus and towards the end of the route, there are 3 passengers {P, Pz2 , P3} and 5 stops SI,S2,S3,S4,Ss, remain. Suppose further that each passenger is inebriated, and is thus is equally likely to get off at any one of the stops (i) We wish to list the set of outcomes in the sample space each of whose outcomes is an ordered triple of all three Sij for l=1,2,3, where Sij means that passenger Pigot off at the stop S a)Write the outcome that P1 and P2 debarked at S4, and P debarked at S. HINT: Its form is (--) where each - is an integer from 1 to 5 b) How many outcomes are in our sample space? Using (i), find the probability that no two of our passengers got off at the same stop (ii) (iii) There will definitely be a fight if exactly two passengers get off at any given stop. (If nobody or 1 passenger gets off at a stop, there is nobody to fight with, and if all three get off at the same stop, one guy will talk the other two out of fighting,.) So what is the probability of a fight occurring? (iv) Finally, suppose there are n stops remaining, and k<n passengers. Answer question (ii) now.

Answer 1

(i) (a) (S14, S24, S31) of outcomes in sample space 5 x 5 x 5 5= 125 passengers got off at the same |(b) Total no. |(ii) Since no two of we have 5 stops i.e. Passenger 1 gets off in any one of 5 stops and then for Passenger 2 we have 4 stops and for Passenger 3 we have remaining 3 stops Hence total no. stop then for first passenger our favourable to the event that of our passengers got off at the same cases stop 5 x 4 x 3 our passengers got off at the same stop no two of Hence required probability that 5 x 4 x 3 two of no S 12 -0.48 52 fight occurring 1-probability that no two of 53 (iii) Probability of 1 0.48 0.52 (iv) Probability of a fight occurring 1-probability that no two of n(n 1 (n k1) passengers got off at the same stop our stop our passengers got off at the same (n 1.(n +1) k-1