SOLUTION: Any help would be greatly appreciated..... The following puzzler was given on the Car Talk radio program on February 19, 2001.* �Three different numbers are chosen at random, an

Question 86634: Any help would be greatly appreciated.....
The following puzzler was given on the Car Talk radio program on February 19, 2001.* �Three different numbers are chosen at random, and one is written on each of three slips of paper. The slips are then placed face down on the table. The objective is to choose the slip upon which is written the largest number. Here are the rules: You can turn over any slip of paper and look at the amount written on it. If for any reason you think this is the largest, you�re done; you keep it. Otherwise you discard it and turn over a second slip. Again, if you think this is the one with the biggest number, you keep that one and the game is over. If you don�t, you discard that one too. . . . The chance of getting the highest number is one in three. Or is it? Is there a strategy by which you can improve the odds?�
The answer to the puzzler is that you can indeed improve the probability of getting the highest number by the following strategy. Pick one of the slips of paper, and after looking at the number, throw it away. Then pick a second slip; if it has a larger number than the first slip, stop. If not, pick the third slip. Find the probability of winning with this strategy.

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Let L represent the largest number
Let M represent the middle number and
Let S represent the smallest number
.
If you were to roll over the numbers one at a time until all three numbers show, there are
six possible outcomes:
.
L then S then M
L then M then S
M then S then L
M then L then S
S then M then L
S then L then M
.
Since all of these are equally likely, they each have a probability of .
.
Notice that two of the outcomes have L as the first draw. That means that the possibility
of L being the first number you turn up is
.
However, your system says that you ignore the first number and proceed. This means that
at this point you have a chance of being wrong and losing.
.
Next look at the possibility that you draw M then S then L. By the rules of your "system" you
discard the M, then draw the S, which being smaller than M, you also discard, and you end
up drawing L on your last turnover. You win. There is a chance of this happening.
.
Next possibility. You draw M then L and then S. By your system you draw M, discard it,
then draw L, and you quit without turning over the third number. You win!. There is a
chance of this outcome.
.
Next possibility. You draw S then M then L. By your system you draw S, discard it, then
draw the M. This being larger than S, you quit ... and you lose because L has not been
drawn.
.
Final possibility. You draw S then L then M. Again by your system, you draw S, discard it,
then draw L. The L being larger than S tells you to quit. You do, and you win! There is
a chance of this happening.
.
Adding it up, you see that you have 3 winning chances [MSL, MLS, SLM] each with a probability
of for a total winning probability of .
.
Your losing chances consist of the two chances that L is the first draw and the chance that
the draw sequence would be SML and you stop at M. This means that the total losing
possibility is
.
So your winning chance using the system is fifty-fifty, a significant improvement
over drawing the first number and quitting.
.
Hope this helps you to understand the problem a little better.
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