A math problem about McDonald's...NEED HELP?

Question

During the summer of 1983, the McDonald's Corporation ran a promotional game. With each purchase, a customer received a game card. Each game card had ten spaces hidden from view. In eight of these spaces, the names of eight different McDonald's products appeared. In one of the remaining two spaces, one of these eight product names appeared again. Thus, nine spaces contained only one possible pairing of product names. The remaining space contained something different; we'll call it an "X". To play, the game card spaces were revealed, 1 at a time, in any order. Play ended when either the pair was uncovered or the X appeared (a loss). Thus, every card could be a "winner". What was the probability of winning this game?

Answer 1

I think firefly's method is correct but difficult to calculate and it looks like xxxx doesn't nuderstand the needed pair.

Begin with a shortened list containing only the two paired (P) items plus the X and write all the ways it can be arranged:

PPX
PXP
XPP

That's it, three ways. It's just a matter of where the X is. The only winning combination is PPX. Now these 3 can be distributed a number of ways among the 10, but for each distribution there are three ways to distribute the pairs and the X and one of the three is a winner. So the odds of winning are simply 1/3.

Answer 2

Here's a hint. Assume that the player always scratches square number 1, then 2, etc., and compute the probability that the "X" is to the right of BOTH of the matching numbers.
That is, say the things behind the squares are: 2 W's, 7 0's and an X.
What is the probability that the X is to the right of both W's?
Well, there is a 10% chance that the X is in the left square.
Also there is a 10% chance that the X is in the second square. Either way, you lose.
If the X is in the third square (10% chance), what is the chance that the Ws are in the first two squares?
etc.

Answer 3

Much as I hate to admit it, I think Pretzels is right. My previous answer failed to take into account the fact that the second pretzel could never be in position 1.