New York is a concrete jungle where dreams are made of To th

New York is a concrete jungle where dreams are made of. To this extent, a young upstart tries to drum up business in three of the tallest buildings in the city. World Trade Center (104 floors), the Empire State Building (86 occupied floors) and the Bank of America Tower (55 floors).

Consider the case where this person enters one of the three buildings randomly and goes to a random floor.

a)Are the building selection and floor selection independent (informationally irrelevant)? Justify your answer using the definition of statistical independence.

b)What is the probability of the businessman winding up on floor 23 of One World Trade Center on a given day? In detail.

c)What is the probability of the businessman winding up on floor 23 of any building on a given day? In detail.

d)If the businessman is on floor 50, what is the probability he is in the Bank of America Tower? In detail.

Solution

a)

No. If they are independent, then the probability of a certain floor is not affected on which building you are in. However, for example, it is impossible to be on the 100th floor if you are on the Bank of America tower. Thus, the probability is affected by the choice of tower; hencem they are NOT independent.

b)

P(floor 23 and OWTC) = P(OWTC) P(floor 23|OWTC)

As buildings are selcted randomly, each building has probability of 1/3. Hence,

P(floor 23 and OWTC) = (1/3) (1/104) = 3/412 = 0.007281553 [answer]

c)

By Bayes\' Rule,

P(floor 23) = P(OWTC) P(floor 23|OWTC) + P(ESB) P(floor 23|ESB) + P(BAT) P(floor 23|BAT)

= (1/3) (1/104) + (1/3)(1/86) + (1/3)(1/55)

= 0.013141703 [answer]

d)

As

P(BAT|floor 50) = P(floor 50 and BAT) / P(floor 50)

Then, as

P(floor 50 and BAT) = P(BAT) P(floor 50|BAT) = (1/3)(1/55) = 1/165

And

P(floor 50) = P(OWTC) P(floor 50|OWTC) + P(ESB) P(floor 50|ESB) + P(BAT) P(floor 50|BAT) = (1/3) (1/104) + (1/3)(1/86) + (1/3)(1/55) = 0.013141703

Then

P(BAT|floor 50) = P(floor 50 and BAT) / P(floor 50) = (1/165)/ 0.013141703

= 0.461173559 [answer]