Question

Question 4: A key ring has 10 keys. Two of these keys are the ones required to open a pair of locks on a safety deposit box. It is virtually impossible to tell the keys apart by just looking at them. If the keys are tried one-by-one, what is the probability that the first three keys tried will not open one of the locks and the fourth will?

Answer 1

There are 10 keys, two of them are the ones that will open the box.

First attempt

In the first chance, there will be 8 keys which will not open the lock and 2 keys which will open the lock

So the probability that the lock will not open in the 1^st attempt = 8/ 10

Second attempt

Once our first attempt is done, we will keep that key aside because we knew that it did not open the lock

So for our second attempt there will be total 9 keys to choose for out of which 7 will not open the lock

So the probability that the lock will not open in the 2^nd attempt = 7/ 9

Third attempt

Once our second attempt is done, we will keep that key aside because we knew that it did not open the lock

So for our third attempt there will be total 8 keys to choose for out of which 6 will not open the lock

So the probability that the lock will not open in the 3^rd attempt = 6/ 8

Fourth attempt

Once our third attempt is done, we will keep that key aside because we knew that it did not open the lock

So for our fourth attempt there will be total 7 keys to choose for out of which 2 will open the lock

So the probability that the key opens one of the locks in the 4^th attempt = 2/ 7

The probability that the first three keys tried will not open one of the locks and the fourth will = (8/10) * (7/9) * (6/8) * (2/7)

= 0.8 * 0.7778 * 0.75 * 0.2857

= 0.1333