Anyone good at maths ????

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Anyone good at maths ????

Postby moonbeam » Aug 9th, '13, 19:39



My local pub has recently started doing play your cards right.

If you're not aware of the game, the rules are simple enough and are as follows:

A full deck of 52 cards is used, the suits are irrelevant as they are all equal and ace is considered high.

2 horizontal rows of cards are dealt, one above the other, with 4 cards in each row, with all 8 cards being dealt face down.

A card is randomly picked from the remainder of the deck and is placed to the left of the bottom row, face up, so there are now 5 cards in the bottom row, the one to the far left is face up and the other 4 are face down.

The aim of the game is to "guess" whether the next card along the row is higher or lower than the one that's just been turned over. So off you go, you've got your 1st card showing and I suppose it makes sense, when "guessing", to go with the better odds. When the 1st card is turned over from the deck, you have the option of keeping the card, or changing it and hoping you get a "good" card, ie one that is very high or low. So when would it be prudent to keep the card and when would you swap ??

If you guess incorrectly at any point, you lose the game.

Anyway, along the bottom row you go and if you make it to the end of the bottom row, that last card is taken and placed alongside the far left card on the top row. You now have the option again of changing this card, or keeping it.

So now you carry on along the top row and if you turn over the last card and guess correctly, then you win.

By the way, if you turn over a pair, you lose.

"Simple" question - what's the expected odds on winning this game.

I'm assuming it would be wise to always go with the odds, so if you get anything from a 2 to a 7, go higher and if you get a 9 to an ace, go lower. An 8 is slap bang in the middle (6 above and 6 below), so I'm not sure how this affects things ..... after taking into consideration any previously turned over cards, if the odds of higher or lower are still equal, maybe just flip a coin to decide ??
When cards have been turned over, you can take these into account when working out the odds of the next card being higher or lower.
I'm assuming that as there are only so many possible permutations of this game, that there would be an expected win/lose ratio.

I'm only asking this, as this has been played in our local pub about 15 times so far and no one has yet won .... the pot is steadily growing .....

Someone care to enlighten me plz ????

QUESTION:
If we can sue McDonalds for making us fat and cigarette companies for giving us cancer; why can't we sue Smirnoff for all the ugly gits we've sh*gged ??
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Re: Anyone good at maths ????

Postby Lawrence » Aug 10th, '13, 21:39

Unfortunatley there's no simple answer here. As each guess and turn of the card changes the probability of the next turn

for reference there are over 57 thousand billion permutations (or 57 quadrillion if you don't believe in proper mathematical terms! (if you're American))

Yeah, in short, your best bet is to just go higher if its lower than an 8 etc....

always remember, just because there are 2 options, does not make it a 50:50 choice :wink:

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Re: Anyone good at maths ????

Postby moonbeam » Aug 11th, '13, 21:01

I worked out the odds of turning over the first card correctly, but this does not take into account the fact that you can change the card if you want:

If first card is a 2 - guess higher and you have a 48 in 51 chance of being correct.
If first card is a 3 - guess higher and you have a 44 in 51 chance of being correct.
If first card is a 4 - guess higher and you have a 40 in 51 chance of being correct.
If first card is a 5 - guess higher and you have a 36 in 51 chance of being correct.
If first card is a 6 - guess higher and you have a 32 in 51 chance of being correct.
If first card is a 7 - guess higher and you have a 28 in 51 chance of being correct.
If first card is a 8 - guess is irrelevant as both will give an equal chance of being correct - 24 in 51.
If first card is a 9 - guess lower and you have a 28 in 51 chance of being correct.
If first card is a 10 - guess lower and you have a 32 in 51 chance of being correct.
If first card is a J - guess lower and you have a 36 in 51 chance of being correct.
If first card is a Q - guess lower and you have a 40 in 51 chance of being correct.
If first card is a K - guess lower and you have a 44 in 51 chance of being correct.
If first card is an A - guess lower and you have a 48 in 51 chance of being correct

Now each of the above has an equal 1 in 13 chance, so I make it that the odds of turning over the first face down card correctly are 160 in 221.

.... I can see that working out the odds of turning the 2nd card over correctly and then the 3rd card, will get exponentially more difficult to calculate .........

QUESTION:
If we can sue McDonalds for making us fat and cigarette companies for giving us cancer; why can't we sue Smirnoff for all the ugly gits we've sh*gged ??
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Re: Anyone good at maths ????

Postby Lawrence » Aug 12th, '13, 08:14

moonbeam wrote:I make it that the odds of turning over the first face down card correctly are 160 in 221.

Cos I'm a maths geek can I see your workings on how you got to this figure?

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Re: Anyone good at maths ????

Postby fiftytwo » Aug 12th, '13, 08:20

I used to quite enjoy the "Hi or Lo" machines on the pier amusements in the 80s when they first arrived - I just consistently picked "Higher" if it was 7 or less and "Lower" if it was 8 or more (iirc they counted Ace as Hi and Lo).

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Re: Anyone good at maths ????

Postby moonbeam » Aug 12th, '13, 08:30

Lawrence wrote:
moonbeam wrote:I make it that the odds of turning over the first face down card correctly are 160 in 221.

Cos I'm a maths geek can I see your workings on how you got to this figure?



moonbeam wrote:I worked out the odds of turning over the first card correctly, but this does not take into account the fact that you can change the card if you want:

If first card is a 2 - guess higher and you have a 48 in 51 chance of being correct.
If first card is a 3 - guess higher and you have a 44 in 51 chance of being correct.
If first card is a 4 - guess higher and you have a 40 in 51 chance of being correct.
If first card is a 5 - guess higher and you have a 36 in 51 chance of being correct.
If first card is a 6 - guess higher and you have a 32 in 51 chance of being correct.
If first card is a 7 - guess higher and you have a 28 in 51 chance of being correct.
If first card is a 8 - guess is irrelevant as both will give an equal chance of being correct - 24 in 51.
If first card is a 9 - guess lower and you have a 28 in 51 chance of being correct.
If first card is a 10 - guess lower and you have a 32 in 51 chance of being correct.
If first card is a J - guess lower and you have a 36 in 51 chance of being correct.
If first card is a Q - guess lower and you have a 40 in 51 chance of being correct.
If first card is a K - guess lower and you have a 44 in 51 chance of being correct.
If first card is an A - guess lower and you have a 48 in 51 chance of being correct

Now each of the above has an equal 1 in 13 chance, so I make it that the odds of turning over the first face down card correctly are 160 in 221.


[48/51 + 44/51 + 40/51 + 36/51 + 32/51 + 28/51 + 24/51 + 28/51 + 32/51 + 36/51 + 40/51 + 44/51 + 48/51]/13 = 160/221

QUESTION:
If we can sue McDonalds for making us fat and cigarette companies for giving us cancer; why can't we sue Smirnoff for all the ugly gits we've sh*gged ??
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Re: Anyone good at maths ????

Postby Lawrence » Aug 12th, '13, 12:42

Ah, OK, I'm with you. so about 75%

Given that the previous outcomes affect the next ones in a[n arguably] negligable way you can simply assume the same maths for the rest of the cards.
so just (166/221)^10 which is about 6%

sounds about right.

Though given you should be taking into account the conditional probability at each stage it's a bit of a vague answer.


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Re: Anyone good at maths ????

Postby Part-Timer » Aug 16th, '13, 21:50

Maybe I have got my maths wrong, but isn't it actually about 10%? Lawrence, I think you might have calculated on the basis of too many cards - there are eight guesses in total, not ten.

But, what is actually being calculated is the chance of the cards following the probability of being higher or lower (which moonbeam sort of acknowledged initially by assuming that the guess is always going to go with the odds). Sometimes, a contradictory guess is better. For example, I saw a couple of episodes of Play Your Cards Right on TV a few weeks back. Twice in those two episodes, I guessed "lower" when the more obvious guess would have been higher. I did this because all the cards revealed before then were highish (nothing lower than a 6, I think). It's therefore possible, in some cases, to improve your chances of winning by paying attention to what's gone before. It's a form of card counting, I suppose!

Two other questions for later:

What are the chances of the "hand" being unwinnable, irrespective of your choices, because there is a pair of cards next to each other? (I actually think this is very difficult to work out because you need to consider the chances of a second card of the same value as another in play coming up somewhere else in the spread and also the ability to swap cards twice in the process throws things out a bit too.)

When should you swap cards? My gut reaction was that it was only worth it if the card was a 7, 8 or 9, but thinking about it some more, I realised it was actually any card from (and including) a 6 to a 10. I based this on the fact that, if the initial card is a 6 or 10, there are 32 better cards in the pack, and 7 cards that are neither better nor worse. You only worsen your position 12 times out of 51. Even if you get a card of a similar level, you actually have very slightly more knowledge about what cards cannot be among the ones in play.

On a very strict mathematical basis, it's worth swapping even if you had a 5 or a Jack (24 better cards, 7 the same, 20 worse), but I don't think many would have the nerve to bin an apparently good card like that.

I suppose that you could also factor in the quality of the card that is worse (getting a 6 when you gave up a 5 isn't too bad, but pulling an 8 leaves you in difficulties), but as a similar consideration applies to the better card you might get, maybe it's actually irrelevant.

This is the first swap only. The second one is a bit more complicated, because you will have some more information about what cards are not left in the pack.

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Re: Anyone good at maths ????

Postby Johnny Wizz » Aug 18th, '13, 10:01

Swap out with a marked deck and the odds are that you will get it right every time. Simples.

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Re: Anyone good at maths ????

Postby Lawrence » Aug 19th, '13, 08:08

Part-Timer wrote:Maybe I have got my maths wrong, but isn't it actually about 10%? Lawrence, I think you might have calculated on the basis of too many cards - there are eight guesses in total, not ten.

You're absolutely right!

I stand by that my maths was correct though! I just used the wrong numbers!

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Re: Anyone good at maths ????

Postby STEVEN52 » Aug 27th, '21, 14:43

Problem here is calculating odds that can vary as you play the game and it progresses.
If you turn a ace, you would obviously bet higher (assuming an ace scores 1) then only 3 new cards turned can lose.(the other aces) the remaining cards are all higher. Therefore its an odds on to win at 12/1 If you turn a 7, the half way number, then it is a 50/50 bet for higher or lower, but biased to lose if another 7 is turned.
You can't pre calculate the odds of winning until you can see all the cards in play and knowing the value of the card you've just turned. As more cards get played, they will change the odds.

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