### Brain Teasers

# Bar Room Toss

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

You are out one night having a few milkshakes when a shadowy figure approaches you. "Wanna play a game?" he inquires. "I'll toss two coins and look at them. If they are both tails then I'll show you both and we'll flip again. On the other hand, if they are not both tails, this is how we'll play. I'll slide one coin across to you. It has to be a head or a tail right? I'll give you $3 if it's a head and you give me $2 if it's a tail. Wanna play?"

Would you play with this stranger and if you decide yes, who will win or is it a fair game?

Would you play with this stranger and if you decide yes, who will win or is it a fair game?

### Answer

If you play you will lose. Outcomes are TT (no play) HH HT or TH. In the last 3 cases the stranger will keep a head, sliding you a tail on two occasions and a head on one occasion. He will win twice as often as you. He will win $4 to your $3 so in the long run you will lose.Hide Answer Show Answer

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## Comments

I got confused. From the question i thought that he was supposed to slide you a random coin.

May 05, 2004

Me too.

C'mon, it's a shady fellow. Do you think he'd slide you a random coin? I give the teaser two enthusiastic thumbs up: one for simplicity and one for practicality. Well done.

My first reaction would have been to run away from the talking shadowy figure, but after it gained my confidence by giving me candy or something, I would listen to its proposition. However, due to the odds against me I still would not have played. Pretty good teaser though!

I got confused. I thought he was going to slide me a random coin, too. Besides, in the first place, I wouldn't even talk to this shady figure. I would run away because I didn't know him.

Well, I was thinking that even when the guy tossed two tails, he could still SAY that he tossed a head and a tail and give me the tail. But, whatever...I guess I'm just more suspicious than everyone else.

Obviously, if he was a shady character, then he would use a double-sided tail coin. That way, the outcomes would be TT, or TH, which results in him always winning!

That's ridiculous. Shady or not, this is supposed to be reasonable. TT=NP. HH=You win, TH=50% HT=50%. Therefore you have a 100%, 50%, or 50% chance of winning with each toss. Those are better than 50% odds, and since you get 50% more money than he does when you win (and you will, more often), it's a good idea to play all night long.

To those who continually doubt, all I can say is sit down with an opponent (who understands the rules of the game) and 2 equal piles of matchsticks. I have convinced many whose logic is somewhat doubtful when their pile of matchsticks slowly transfers itself to mine. What is really nice about this teaser is that common sense tells you you will win. Probability tells us that you will lose and if you actually play the game as per the rules (assuming th character who is tossing the coins is trying to win) then you can confirm that you will definitely lose!

I am the Omega's comment is rubbish. TT is no play. The chance of getting HH is 1 in 4. I think therefore the chance of getting at least one tail is 3 in 4. I believe the game is even more weighted aagainst you than the compiler says.

Don't be stupid, the compiler is right you will lose 2 out of 3 games and therefore can expect lose $1 for every 3 games played on average. There is not a 1/4 chance of getting HH there is a 1/3 chance as all TT results are disgarded so when calculating expected payouts you need to act like that result is never an option.

Just to point out to Omega why exactly it is not a good game to play, the chance of winning when the result is TH or HT is not 50% as you say but infact 0%. This is because the character has already looked at the two coins and is not going to be sliding you a coin that results in himself losing when he has the option of winning available is he.

Mar 06, 2005

veddy veddy goot.

I decided that I wouldn't want to play this game mainly because I couldn't figure out the odds.

can i say this. I HATE PROBABILITY TEASERS no offense to yours in paticular but they always strike up heated competition and i never can agree

It must be a good teaser, because Marilyn vos Savant's editor let her present the same puzzle in the nineties. (The participants in her teaser were feminine.)

Interesting. I devised this teaser originally I think from a problem suggested at a computer science summer school back in the eighties. We were looking at a problem for some high school students to program on a computer as a simulation using a random number generator. I have modeled this in may different ways using 2 black cards and 2 red card etc. It is a classic problem in conditional probability very similar to others of mine and some of Cathal McCabes on this site. PS I've never heard of Marilyn.

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