Optimal strategy for The Price is Right game "Punch A Bunch"

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There are $64,000 in prizes on the board. With fifty slots occupied, the average value is $1280. The expected value of a pick is a little trickier due to the second chance cards, but let's say that they add another four shots at the "average" prize, so they are each worth $1280 more than their face value. This is an additional $5120, so now we can say that a pick is worth (64,000+5120)/50, or $1382.

So a quick rule of thumb might be to continue picking, as long as picks were available, until one of the top five prizes was obtained. The EV of a pick would rise as subsequent picks were made, because after the initial low-value picks, the average of the remaining picks would rise.

The above notwithstanding, I would expect most contestants to stop and take the prize if they got $1,000, and maybe even $500.

mkl, you're mistaken in your calculation of the expected value of the first pick. This is the correct formula:

(64,000 + 4x) / 50 = x,

Where x is the expected value of the first pick (the 4x represents the 4 second chances.)

This gives EV=x=$1,391.

Since there are only a maximum of 4 picks, I know you should never settle for a prize less than $5,000.

This is illustrated by this simple example:

Suppose on your first punch you hit $1,000. Do you keep it or punch again? If you keep it you get $1,000. If you punch again, your next pick has an EV of (63,000 + 4x)/49 = x, and we get an EV = x = $1,431 which is greater than the $1,000. So you should punch again.

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Quote: cardshark

mkl, you're mistaken in your calculation of the expected value of the first pick. This is the correct formula:

(64,000 + 4x) / 50 = x,

Where x is the expected value of the first pick (the 4x represents the 4 second chances.)

This gives EV=x=$1,391.

Since there are only a maximum of 4 picks, I know you should never settle for a prize less than $5,000.

This is illustrated by this simple example:

Suppose on your first punch you hit $1,000. Do you keep it or punch again? If you keep it you get $1,000. If you punch again, your next pick has an EV of (63,000 + 4x)/49 = x, and we get an EV = x = $1,431 which is greater than the $1,000. So you should punch again.

You reached the same basic conclusion as I did--I think the difference between my calculation and yours is a slight disagreement over the effect of the second chance rule.

I come up with a slightly different answer. I assume you'll always repunch when you get another chance. Therefore, I remove these four squares and come up with a punch EV of $1372. Same basic conclusion as the above posters. Keep hitting until you have a prize of more than $1000.

Of course someone might want to keep $1000 if that money means alot to them.

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For those who are confused by "second chance" cards:
Say you only get 2 punches because you sucked at the higher/lower portion of the game. You then punch 2 compartments (#1 and #2). Drew will then pull out the #1. if it is one of the 4 second chance cards, you will then pick 1 more compartment(#3). Drew will then add #1 and #3 to get your first offer. You can then reject or accept this offer.

Quote: miplet

For those who are confused by "second chance" cards:
Say you only get 2 punches because you sucked at the higher/lower portion of the game. You then punch 2 compartments (#1 and #2). Drew will then pull out the #1. if it is one of the 4 second chance cards, you will then pick 1 more compartment(#3). Drew will then add #1 and #3 to get your first offer. You can then reject or accept this offer.

AIEEEEEEE! Drew Carey!!! You just ruined my mental image of the game, which still had Smiling Bob Barker in it!!

What happens if the player gets two second chance cards? I assumed he got to take the sum of all three monetary cards as the prize for that round. If that assumption is correct, then I show the average value of the first round is $1390.89. I had a different solution than Cardshark, which considered exactly four second chances cards, and the ranges at which they were placed. Cardshark assumed they could be anywhere, and each hole had an independent 4/50 chance of having one. However, it obviously works out the same.

I find the strategy of the game is to always trade with less than $5,000, regardless of how many rounds you have left.