In: Statistics and Probability
We have n people and K floors: what is the chance that 1)all people get off in the smae floor? 2)at least one in evey floor?3)P( floor 3 is empty)?4)p(2 people get out in the first floor and 3 people or mpre get off in the second floor)
Answer 1: All people get off on the same floor:
Probability of choosing any single floor = 1/k
Since there are n people so Required probability = KC1 x (1/k)^n
2) Atleast one get on each floor = 1-(one floor is empty + 2 floors are empty .....k floors are empty)
= 1- (Kc1 x (1-1/k)^n + Kc2 x (1- 1/k)^n.............+Kck x (1-1/k)^n)
= 1- (1-1/k)^n X ((2^k)) ( 1 + 1
= ( k^n - ((k-1)^n X (2^k )))/ k^n
Answer 3: Floor 3 is empty = Kc1 x Q^n ( Q= 1- P = 1- 1/k )
= Kc1 x (1-1/k)^n
Answer 4: p(2 people get out in the first floor and 3 people or more get off in the second floor)=
p(2 people get out in the first floor) x p(3 people or mpre get off in the second floor)
p(2 people get out in the first floor)= Kc1 x (1/k)^2
p(3 people or more get off in the second floor)= K-1c1 x ((1/k-1)^3 + (1/k-1)^4.................+ (1/k-1)^n-2 )
=K-1c1 x ( 1- ( (1/k-1)^1 + (1/k-1)^2 )
Therefore, p(2 people get out in the first floor) x p(3 people or mpre get off in the second floor)=
Kc1 x (1/k)^2 x K-1c1 x ( 1- ( (1/k-1)^1 + (1/k-1)^2 )