Sweepstakes ... probability

Is the car key Sweepstakes a fair contest?

A national car manufacturer had a nation wide promotion.
They selected six finalists.

During a commercial break of a sporting contest each of the six
contestants were to choose one of six keys, sitting on a table.

Only one of the keys would start the car.

The first person picked a key, tried to start the car, and failed.

The next person picked a second key and it started the car.

Immediately people watching declared it wasn't fair.

The complaints:

• The last four people didn't get a chance to select a key, therefore they never had a chance to win.
• The first person should have put the key back after they found it didn't work, then all the keys should have been mixed up again, and then the second person should try to pick the key from the six like the first person did.

1. Was the car key contest fair?
2. Did each person have as much of a chance to win as the other?
3. If not which of the above changes would make it fair?
4. Was picking second the best time to get to pick and try a key ?
5. If not, then, when was the best time to pick for the best chance to win?
6. If the contest organizers would have had each person pick a key before any tried them in the car, would that have changed the viewers perceptions of the contest as being fair.

Hint: Prove that no position has an advantage. If the first person had a probability of 1:6 of getting the key that started the car, did the next person have a probability of 1:5? Then 1:4...? What were the the chances that two people would win the car? What were the chances that less than one person would win the car? Was the second person's first pick really a first pick?

Discussion: