You are given seven keys, and you know that one of them opens a treasure chest.

Question

2. You are given seven keys, and you know that one of them opens a treasure chest. 

    You try one key after another, in random order. 

What is the chance that you will open the chest on the first try? 

Second try? Third try? 

What pattern do you see? Can you justify the pattern that you see?

Answer 1

First try: 1/7 = A

Second try (assuming you take the key tried in the first out of contention): (1 - A) * 1/6 = B

The (1 - A) term is the likelihood the your first try failed. The (1/6) terms comes from the fact that six unused keys remain and one will open the lock.

Third try: (1 - A - B) * 1/5 = C

Here we multiply the probability of getting past try 1 then try 2 before picking one of the 5 remaining keys.

Fourth try: (1 - A - B - C) * 1/4 = D

Fifth try: (1 - A - B - C - D) * 1/3 = E

Six try: (1 - A - B - C - D - E) * 1/2 = F

Last try: (1 - A - B - C - D - E - F) * 1 = G

In general, the probability of opening the lock with the nth key is equal to 1 over the number of unused keys times (1 - probability of success on any prior try).

If you check, you'll see that the probability is always 1/7.