[quote name=\'JasonA1\' date=\'Nov 6 2005, 12:00 PM\']
How many people with the ego to want to be on a game show are going to do that?
Very true. A lot of times on shows contestants don't take the necessary step back of "hey, I could have a guaranteed $20,000. I only have a slight chance to win more" that causes game stops. Just look at Greed. Or even some instances on The Price is Right - most notably in my mind, the reams of players who give up $1,000 in Punch-a-Bunch.
-Jason
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Sorry to be pedantic, but you'd be very surprised to know that if you get four punches in Punch-a-Bunch, and the first ist $1000, it actually is in your best interest to give it back: the probabilities favor winning at least $1000 anyhow.
After the first pull, there are 10 each of the $50, $100, $250, and $500 cards remaining, as well as 4 $1000, 3 $5000, and 2 $10000 cards. The probability of not getting any of the $1K or more cards is:
40/49 * 39/48 * 38/47 = 53.626%, which means that the other 46+ percent of the time you
will get a $1K or better card, with the probabilities as follows:
Exactly one $1K or better card: 40/49 * 39/48 * 9/47 * 3 (since the card can be any of the second, third, or fourth punches) = 38.102%
Two $1K or better cards: 40/49 * 9/48 * 8/47 * 3 = 7.816%
All three punches $1K or better: 9/49 * 8/48 * 7/47 = 0.455%
However, there are those 5 pesky little "second chance cards". There's the possibility of getting one of those, and then punching a $1K card. (Didn't that just happen last week?) The probability of this happening is 40/49 * 39/48 * 5/47 * 9/46 * 3 = 4.141%.
This, added to the 46.373% of getting a straight $1K without the second chance, already puts you over the 50% mark.
(Also, thanks to the second chance cards, the average expected value of any one punch is over $1K....)