Question

Suppose that 12 people enter an elevator on the 1st floor of a 24 floor building. Assume that all 12 independently pick a floor (above the first) randomly to get off on. What is the expected number offloors no one gets off on?

Answer 1

Answer:

∑E(x$_{i}$) = 13.49167 floors

Step-by-step explanation:

The expected number of floors no one get off = ∑E(x$_{i}$) where i is from 0 to 23

and E(x$_{i}$) = ∑x$_{i}$P(x$_{i}$)

here x$_{i}$ is the indicator of floor where no one gets off, its value is 0 when  atleast one person get off on its floor and 1 when when no one gets off.

Now,

P(x$_{i}$=1) = (22/23)¹²

P(x$_{i}$=0) = [1-(22/23)¹²]

Now,

E(x$_{i}$) = ∑x$_{i}$P(x$_{i}$) = 0* [1-(22/23)¹²] + 1*(22/23)¹² =0.586594704

For total number of floors where no one gets off

∑E(x$_{i}$) = E(x₁)+E(x₂)+E(x₃)........................+E(x₂₃)

∑E(x$_{i}$) = 23*0.586594704

∑E(x$_{i}$) = 13.49167 floors