### Brain Teasers

# Playing the Penny

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

You are in a bar having a drink with an old friend when he proposes a wager.

"Want to play a game?" he asks.

"Sure, why not?" you reply.

"Ok, here's how it works. You choose three possible outcomes of a coin toss, either HHH, TTT, HHT or whatever. I will do likewise. I will then start flipping the coin continuously until either one of our combinations comes up. The person whose combination comes up first is the winner. And to prove I'm not the cheating little weasel you're always making me out to be, I'll even let you go first so you have more combinations to choose from. So how about it? Is $10.00 a fair bet?"

You know that your friend is a skilled trickster and usually has a trick or two up his sleeve but maybe he's being honest this time. Maybe this is a fair bet. While you try and think of which combination is most likely to come up first, you suddenly hit upon a strategy which will be immensely beneficial to you. What is it?

"Want to play a game?" he asks.

"Sure, why not?" you reply.

"Ok, here's how it works. You choose three possible outcomes of a coin toss, either HHH, TTT, HHT or whatever. I will do likewise. I will then start flipping the coin continuously until either one of our combinations comes up. The person whose combination comes up first is the winner. And to prove I'm not the cheating little weasel you're always making me out to be, I'll even let you go first so you have more combinations to choose from. So how about it? Is $10.00 a fair bet?"

You know that your friend is a skilled trickster and usually has a trick or two up his sleeve but maybe he's being honest this time. Maybe this is a fair bet. While you try and think of which combination is most likely to come up first, you suddenly hit upon a strategy which will be immensely beneficial to you. What is it?

### Hint

Think what would be most likely to happen if you chose HHH, would this be a good decision?### Answer

The answer is to let your friend go first. This puzzle is based on an old game/scam called Penny Ante. No matter what you picked, your friend would be able to come up with a combination which would be more likely to beat yours. For example, if you were to choose HHH, then unless HHH was the first combination to come up you would eventually lose since as soon as a Tails came up, the combination THH would inevitably come up before HHH. The basic formula you can use for working out which combination you should choose is as follows. Simply take his combination (eg. HHT) take the last term in his combination, put it at the front (in this case making THH) and your combination will be more likely to come up first. Try it on your friends!Hide Hint Show Hint Hide Answer Show Answer

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## Comments

I see it now. "flipping the coin continuously" is the key to this. I initially incorrectly read this as he would flip three coins see if they matched anyones guesses and then start fresh and flip the coin three more times, etc. That is much different than what you stated. Good teaser.

I'm still not sure of the answer. If the other person chose HHT, THT and TTH, what possible combination could you beat him with?

Great question!

I get it! This is a great teaser! However, I recommend you include a more detailed explanation, as it took me a very long time to run through all the combinations in my head.

I think the explanation for the second person's strategy is easier to understand if you look at it from the standpoint that the second person controls which two combinations aren't chosen. Â¾ of the time the game ends on the first three flips of the coin with no advantage to either person, so it's only the probabilities following one of the two remaining sequences that are important. The strategy is then to leave open sequences that give the other person the worst chance of completing a sequence.

Depending on the choice of the first person, the best probability the second person can achieve varies from 1/2 to 5/8.

HHT and TTH with either THT or HTH are the only the two choices that guarantee the first player at least a 50% chance of winning. HHT and TTH are the best choices for the first person because they force the second person to take HHH and TTT which only neutralize each of the first person's choices without giving an advantage to the second person. (HHT forces HHH because HHH only leads into HHT and itself, meaning that if HHH is one of the open sequences then it will always be a winner for whoever has HHT.) If the second person takes TTT and HHH then both HTT and THH are neutral as open sequences (each person has a 50% chance of winning). That means the first person only needs to worry about THT and HTH as the open sequences. Since THT and HTH lead into each other, taking either one forces the second person to take the other one, leaving the two neutral sequences HTT and THH.

Depending on the choice of the first person, the best probability the second person can achieve varies from 1/2 to 5/8.

HHT and TTH with either THT or HTH are the only the two choices that guarantee the first player at least a 50% chance of winning. HHT and TTH are the best choices for the first person because they force the second person to take HHH and TTT which only neutralize each of the first person's choices without giving an advantage to the second person. (HHT forces HHH because HHH only leads into HHT and itself, meaning that if HHH is one of the open sequences then it will always be a winner for whoever has HHT.) If the second person takes TTT and HHH then both HTT and THH are neutral as open sequences (each person has a 50% chance of winning). That means the first person only needs to worry about THT and HTH as the open sequences. Since THT and HTH lead into each other, taking either one forces the second person to take the other one, leaving the two neutral sequences HTT and THH.

Great - a genuine teaser that goes beyond intuition and the math classroom!

However, the solutions seem to overlook the question scenario in which 'you' will choose first. This makes me suspect that the friend knows the trick. There is therefore NO strategy that could be of 'Immense Benefit' to you - 50:50 or at best 5/8 is not that good.

Best strategy is to follow your grandma's advice - avoid the gambling dens and stay sober!

However, the solutions seem to overlook the question scenario in which 'you' will choose first. This makes me suspect that the friend knows the trick. There is therefore NO strategy that could be of 'Immense Benefit' to you - 50:50 or at best 5/8 is not that good.

Best strategy is to follow your grandma's advice - avoid the gambling dens and stay sober!

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