A box contains five keys, only one of which will open a lock. Keys are randomly selected and tried, one at a time, until the lock is opened (keys that do not work are discarded before another is tried). Let Y be the number of the trial on which the lock is opened.
Find the probability function for Y.
My work: Obviously, the probability of the first key working is 1/5. To find the probability of the first key not working and the second key working, I multiplied (4/5)*(1/4). Similarly, the probability of the first and second keys not working and the third key working is (4/5)*(3/4)*(1/3). Am I on the right track? The probability ends up being 1/5 for each key, and I was concerned because I thought it should be different for each one.
Thanks for your help!
Re: Probability Function
Think of it this way: line up the keys in a random order. What is the probability that the first key is the correct key? The second key? Etc. It is 1/5 for each position. Now, try the keys in the order you laid them out. Whatever position the key was in, that is the trial number when the door will open. So, since the order you laid them out was random, they must have the same probability of occurring.