sian. life in the navy is so unpredictable. yesterday evening was so busy crafting and drafting a "opening address" for today's event (which the speaker capriciously chose not to use) and today am so boh liao to do the following calculations...
the numbers game - on channel 8, tues, 8pm.
in the final round (i only saw the final round, hee), there were nine cards labelled 1 to 9. the card '5' was picked out while the rest were covered and mixed up. the contestant was then to pick out one card and the aim was to guess correctly if the chosen card is bigger or smaller than ‘5’. if the guess is right, his winnings will be doubled. if not, the winnings will go to zero.
the numbers game - on channel 8, tues, 8pm.
in the final round (i only saw the final round, hee), there were nine cards labelled 1 to 9. the card '5' was picked out while the rest were covered and mixed up. the contestant was then to pick out one card and the aim was to guess correctly if the chosen card is bigger or smaller than ‘5’. if the guess is right, his winnings will be doubled. if not, the winnings will go to zero.
of cos, it was just that. then it'll be boring. so, there’s this guru who’ll thru some magical calculations and offer the contestant to open one of the other eight uncovered and unchosen cards for a certain amount of the contestant winnings.
example: for today, zhenrong chose a card (which of he didnt know before hand) and guessed that it was higher than ‘5’. then, the guru offered him $300 to open one of the eight unchosen cards. he took the offer and opened a ‘3’, which seem to imply that the card is more likely to be higher than ‘5’. Then the guru offered $350 for another card and he opened a ‘6’. Oops. Now it’s equally likely and he through some instinct decided to guess smaller than ‘5. Next, the guru offered $200 for another card and zhenrong decided to not open anymore. and fortunately the chosen card was ‘1’.
anyway... was boh liao and decided to do some calculations and figure out some strategy. made the following assumptions.
1. the contestant is not some gambler god who can memorise the positions of the cards. so the opening and choosing of the cards is totally random.
2. this one is debatable. i assumed the guru will make an offer such that the expected winnings without opening a card is the same as the expected winnings after opening a card. sounds complicated. for simple reasons, if the offer is too good, it’ll be to the disadvantage for the guru/banker. if the offer is too lousy, the contestant will not take the bait.
for example, if there are only two uncovered cards ‘1’ and ‘9’ left (one of which is the chosen one) and the contestant has only $100 left. then without opening, the expected winnings is (1/2)($200)+(1/2)($0)=$100. if the contestant opens the unchosen card, he’ll know the chosen one. so, my assumption is that the guru will offer $50. and the contestant will take back 2($100-$50)=$100.
as i said, this assumption is debatable. during the game, the guru never followed this offering system. and this assumption also needs a prerequisite that the contestants will do the necessary calculations. haha.
my conclusion.
case 1: the number of uncovered cards bigger than 5 = the number of uncovered cards smaller than 5. then open another card only if the offer the guru makes is less than (your winnings/number of uncovered cards). by right, the guru should make the offer = (your winnings/number of uncovered cards), following assumption 2.
case 2: the number of uncovered cards bigger than 5 and the number of uncovered cards smaller than 5 are not equal. this conclusion is quite surprising. do not take any offer as any positive offer will only lower the expected winnings. (of cos, if offer free, then take lor...)
disclaimer.
the expected winnings is defined in the statistics sense. that is, the strategy should be work in many many games. but, in the game, i saw, zhenrong nv followed the strategy, but he still won. just like any lottery thing lah. everyone is expected to lose, but there will always be the exception or outlier (which everyone hopes to be).
i also assumed the offer is to open one card. perhaps for two or more cards, the strategy may differ. and most likely, channel 8 will alter the "offer" contents to keep the audience interested.
my calculations may be totally wrong.... haha
example: for today, zhenrong chose a card (which of he didnt know before hand) and guessed that it was higher than ‘5’. then, the guru offered him $300 to open one of the eight unchosen cards. he took the offer and opened a ‘3’, which seem to imply that the card is more likely to be higher than ‘5’. Then the guru offered $350 for another card and he opened a ‘6’. Oops. Now it’s equally likely and he through some instinct decided to guess smaller than ‘5. Next, the guru offered $200 for another card and zhenrong decided to not open anymore. and fortunately the chosen card was ‘1’.
anyway... was boh liao and decided to do some calculations and figure out some strategy. made the following assumptions.
1. the contestant is not some gambler god who can memorise the positions of the cards. so the opening and choosing of the cards is totally random.
2. this one is debatable. i assumed the guru will make an offer such that the expected winnings without opening a card is the same as the expected winnings after opening a card. sounds complicated. for simple reasons, if the offer is too good, it’ll be to the disadvantage for the guru/banker. if the offer is too lousy, the contestant will not take the bait.
for example, if there are only two uncovered cards ‘1’ and ‘9’ left (one of which is the chosen one) and the contestant has only $100 left. then without opening, the expected winnings is (1/2)($200)+(1/2)($0)=$100. if the contestant opens the unchosen card, he’ll know the chosen one. so, my assumption is that the guru will offer $50. and the contestant will take back 2($100-$50)=$100.
as i said, this assumption is debatable. during the game, the guru never followed this offering system. and this assumption also needs a prerequisite that the contestants will do the necessary calculations. haha.
my conclusion.
case 1: the number of uncovered cards bigger than 5 = the number of uncovered cards smaller than 5. then open another card only if the offer the guru makes is less than (your winnings/number of uncovered cards). by right, the guru should make the offer = (your winnings/number of uncovered cards), following assumption 2.
case 2: the number of uncovered cards bigger than 5 and the number of uncovered cards smaller than 5 are not equal. this conclusion is quite surprising. do not take any offer as any positive offer will only lower the expected winnings. (of cos, if offer free, then take lor...)
disclaimer.
the expected winnings is defined in the statistics sense. that is, the strategy should be work in many many games. but, in the game, i saw, zhenrong nv followed the strategy, but he still won. just like any lottery thing lah. everyone is expected to lose, but there will always be the exception or outlier (which everyone hopes to be).
i also assumed the offer is to open one card. perhaps for two or more cards, the strategy may differ. and most likely, channel 8 will alter the "offer" contents to keep the audience interested.
my calculations may be totally wrong.... haha
2 comments:
wah quite cheem....ur blog not very reader-friendly leh..hahha oni for mathematics pple ah. haha
HUH? *blur look*
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