You just rented a large house and the realtor gave you five keys, one for the fr

Question

You just rented a large house and the realtor gave you five keys, one for the front door and the other four for each of the four side and back doors of the house. Unfortunately, all keys look identical, so to open the front door, you are forced to try them at random. Find the mean and the variance of the number of trials you will need to open the door, under the following alternative assumptions: (a) after an unsuccessful trial, you mark the corresponding key so that you never try it again, or (b) at each trial, you are equally likely to choose any key.

Explanation / Answer

a) for probability to get it opened in 1 trail =P(X=1) =1/5 =0.2

probability to get it opened in 2nd trail=P(X=2)=P( not opened in first and opened in 2nd)=(4/5)*(1/4) =1/5

similarly P(X=3) =P(X=4)=P(X=5) =1/5=0.2

from above information:

mean =3

and variance =2

b)here it is geometric distribution with parameter p =P( probability of finding the key ) =1/5 =0.2

hence mean =1/p=1/0.2 =5

and variance =(1-p)/p2 =(1-0.2)/0.22 =20

x p(x) xP(x) x2P(X) (x-)2 (x-)2P(x) 1 0.2000 0.200 0.200 4.000 0.800 2 0.2000 0.400 0.800 1.000 0.200 3 0.2000 0.600 1.800 0.000 0.000 4 0.2000 0.800 3.200 1.000 0.200 5 0.2000 1.000 5.000 4.000 0.800 total 1 = 3.00 11.000 10.000 2= 2.0000

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