Braingle: 'Two Coins' Brain Teaser

Two Coins

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

Puzzle ID:	#29228
Fun:	(2.66)
Difficulty:	(1.81)
Category:	Probability
Submitted By:	lips
Corrected By:	Winner4600

Suppose I flip two coins without letting you see the outcome, and I tell you that at least one of the coins came up heads. What is the probability that the other coin is also heads?

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Comments

Woden Mar 21, 2006	I can't agree with this. The possible outcomes of two coin flips are HH, HT, TH, TT. Since the first flip is known to be H, the possible outcomes are now HT and HH. You can't claim TH to be valid any more. It turns into a single coin flip.
tsimkin Mar 21, 2006	He didn't say that the first was heads, he said at least one was heads. There are indeed three equally-likely ways to get "at least one" with heads, and only one of these has two heads. The answer is 1/3.
cms271828 Mar 21, 2006	This is analogous to the dogs question, so it shouldn't really be here. If you don't know which of the coins is a head, the chance of getting 2 heads is 1/3. I've tested it with a computer program, and the result just keeps getting closer to 0.33333 the more experiments are done. You can use a tree driagram too. Basically after there are 4 options HH,HT,TH,TT, each with a 1/4 chance of occuring, but since TT must be ruled out, we only have 3 equal options HH,HT,TH, each with a 1/3 chance.
pating Mar 22, 2006	i think the answer would have to be 1/4. as you have said, there are 4 possible combinations. and knowing that at least one coin turned up heads doesnt change the probability of getting HH in flipping 2 coins. we cant just eliminate TT from the equation bcoz before you flipped the coins, TT has 1/4 chance of turning up. and it cant be 1/2 either (no, it doesnt turn into a single coin flip), bcoz its like saying that the first coin is sure to turned up heads. but if you said right from the start that the coins are somehow biased that at least 1 coin will always turned up heads, then i would agree that the probability of getting HH is 1/3. otherwise if getting HH, HT, TH or TT have equal chances, then it should be 1/4. (the 1st coin has 1/2 chance of turning up H, and the other have 1/2 chance also. so 1/2 x 1/2 = 1/4!) (thats what i learned from joining math contests back in elementary and high school . if i remembered correctly, this is just a trick question! it makes you think of other answers when in fact, the answer is quite simple.)
jj_is_cool Mar 24, 2006	wow, i didnt follow any of those comments! still good teaser
Jimbo Mar 24, 2006	[1] This is a duplicate of the teaser "Bar Room Toss". [2] It should be stated "What is the probability that they are both heads?" because once you say "The other one" it implies that you are indicating that a particular coin is a head. In that case the answer would be 1/2. [3] Provided the wording is changed to "both heads" instead of "the other one is a head" then the answer is 1/3.
OldChinaHand Mar 25, 2006	So much ado about a simple textbook case in simple probability 101; it is 1 in 3 chances.
Jimbo Mar 27, 2006	I take back point [2]. In this context, "The other one is a head" and "both heads" are equivalent. Sorry!
brainjuice Mar 27, 2006	why not 1/2? i think we can assume that there's only one coin which need to be tossed.. i dunt really undrstand (i'm a beginner).. can anybody explain? thx.
cms271828 Mar 28, 2006	Yeh, because basically you don't know which of the 2 coins is a head to start with. There are 3 equally likely options: HH, HT, TH
TEASEME315 Mar 30, 2006	THIS WAS AN EXCELLENT TEASER, I SAY THIS BECAUSE ONE: IT STARTED A WHOLE WORLD OF DISCUSSION AND THE ANSWER IS VERY SIMPLE AND TWO: I LOVE PROBABILITY. HOWEVER, THE ANSWER IS 1/3(IN MY OPINION.)
mmmcla01 Mar 30, 2006	It would be 1/2 because the events are independent of eachother. One occurence does not influence the other
PatH Apr 04, 2006	Okay, here is a clear explanation why 1/3 is correct. Lets start from the beginning. The chance of getting two heads (HH) if you flip two coins is 1/4. This is true because you have four possible outcomes HH, HT, TH, TT. Mathematically, if you have a 1/2 chance of each coin landing H, then 1/2 * 1/2 = 1/4 The question posed is this: What is the chance of having two heads (HH) if I tell you that the outcome is NOT two tails (TT)? [This is the EXACT same question as the teaser, worded differently] This now means we have three possibilities HH, HT, TH. The chances of getting HH, of these three possibilities, is 1/3. Basically what is happening here, is 1/4 of the data is being thrown out, that is all occurences of TT are not counted, bringing the total possibilities to 3. This is NOT the same as the following question: If I have a coin that is heads, what is the chance of flipping another coin and having two heads? In that case, the chance would be 1/2. The difference here is simple, you are eliminating the chance of coin A landing tails, leaving the possibilities HH and HT. The main point of confusion here is the fact that HT and TH are NOT equal. These are two distinct events, even though they appear the same. Try this at home using a dime and a penny, keeping track of what each coin does independently. Hope this helps.
(user deleted) Apr 06, 2006	I totally agree with the comment given by PATH, since, earlier, i was also confused with the same stuff. A nice teaser
Atropus Apr 07, 2006	..... and if you don't believe it.. test it.
Swordoffury1392 Apr 09, 2006	The answer is definately one half. The basic problem in the concept of the teaser is that HT and TH are the same exact thing. If one of the coins is head, then the other coin being heads or tails is unaffected. There is still, no matter what, a 1/2 chance of the other being heads. This teaser is not really s teaser but a paradox that seems to confuse people with misinformation such as the teaser with the three men going into a hotel and there being a missing dollar. Sorry...
PatH Apr 10, 2006	Swordoffury, Please take another moment to think about your logic. HT and TH are actually not the same thing. Ask yourself the following questions: What is the chance of getting HH? 1/2 x 1/2 = 1/4 What is the chance of getting TT? 1/2 x 1/2 = 1/4 What is the chance of getting one heads and one tails? This is just 100% - 25% - 25% (from above) 1 - 1/4 - 1/4 = 1/2. As you can see, you are twice as likely to get heads and tails combined, because there are two ways to accomplish it. Use Excel to generate random 1s and 0s in two columns to model this situation. It is a great proof by experiment.
dimashkieh Apr 13, 2006	I disagree with the answer. The probability would be the same if flipping one coin which is 1/2...
GeniusGod Apr 24, 2006	The answer would be 1/2 for the coin to land on heads. (Two events) It would be 1/3 for both coins to land on heads (one event) You need to reword your question
keosborne Apr 28, 2006	Thanks for the explanation, PatH... it was very helpful. I think I got it now!
Cridol May 02, 2006	1 out of 2
PatH May 04, 2006	Cridol, Your comment is snotty and ignorant, provide some explanation for your claim if you want to be taken seriously.
rock57 May 16, 2006	This is similar to the following: A prize is behind 1 of 3 doors. After you pick a door, the host opens 1 of the other doors, which does NOT contain the prize. If you are allowed to change your choice to the other unopened door, should you do so? The answer is YES. By changing, you have a 1/2 chance of winning. If not, then it is 1/3. (from Ask Marilyn in Parade.)
tsimkin May 17, 2006	rock57 -- two comments. (1) In the games show example you use, the probability of winning if you switch is 2/3, and if you stay is 1/3. Otherwise, the sum of the probabilities would be less than 1, which makes no sense. (2) This is not really the same issue. In the game show, there is asymmetric information which allows the host to show you "the goat"; here, there is just the elimination of the TT case, but not the same shift from a priori probabilities.
rock57 May 17, 2006	tsimkin: Of course you are correct. (I had a brain cramp)
gromney May 22, 2006	The solution 1/3 is correct. There are only two coins being tossed. Since there are only four possibilities, write them down on a piece of paper. Cross out the two tails possibility, since we are told at least one coin is heads. See what you have left. Walk through the logic and come up with the answer.
scotttie May 23, 2006	This conversation has been amusing to read. Both people on both sides are so sincerely addiment that they are correct. Basically you both could be right if the wording was slightly different in several various ways in 3 areas, the tense of the questions and the detail of knowledge of the outcome and the level of detail needed to be seen in result given. I challenge you to toss 2 coins (and repeat this exercise 20+ times). If at least one of the coins comes up Heads (as the riddle states has happened, we officially know at least one of the coins is a Head) then you are able to use the result in this experiment Now, mark down if the face of 'THE OTHER' coin was heads or tails. This is a practical example that does what the question states. You will find the result. The result will surprise half of you! I haven't time to explain tonight why both arguments are so strong but one is incorrect, maybe later in the week if people still doubt this experiment. Probability is so simple but can easily deceive with its presentation. Scotttie
udoboy May 26, 2006	I agree with Sword of Fury, kind-of. HT and TH are the same thing, but they don't then comprise 1/2 of the remaining chances, but 2/3 of the remaining chances. So the 1/3 solution is still correct.
Vudluxi May 29, 2006	This seemed like such a nice, and correctly worded, teaser to create such controversy. For 2 coin tosses there are four outcomes. 3 of these include at least one head. We are told there is at least one head. Only one of the outcomes has a second head. 1 in 3 is the correct and only answer. The 2 tosses had already been made. We are not asked if a head has just been tossed what's the probablilty of a second head. What's the probablilty of everyone getting their heads around this? Very slim
JeffJ99 Jun 04, 2006	Seemed like 1/2 after the fisrt read, but 1/3 is certainly the correct answer.
crumbbum Jun 17, 2006	The problem with 1/3 is that one coin toss has already been decided. The prompot told us "that at least one of the coins came up heads". Thus either toss 1 or toss 2 is heads before we start the problem, so what is the probability that "the other coin is also heads". To me, the language suggests we're considering only 1 flip. The "also", especially, implies that one coin is already set as heads and we're dealing with the other coin.
mmmcla01 Jun 20, 2006	By changing the wording to: "What is the probability that both coins came up heads" you would eliminate a lot of confusion.....however it is true that the answer is 1/3 because you're not looking at two separate events but rather one sample space that includes two events.
crumbbum Jul 01, 2006	After reading this again I'm convinced that the way you worded it is wrong, and that it should be 1/2.
grungy49 Jul 20, 2006	I disagree with this. I think it should be a 1 in 2 chance. The question is worded poorly. 'Suppose I flip two coins without letting you see the outcome, and I tell you that at least one of the coins came up heads. What is the probability that the other coin is also heads?' It doesn't matter that the first coin was heads. That doesn't have anything to do with the equation. The question is: What is the probability that the other coin will land on heads. I understand what the author means with the answer explanation, but I think you should rethink how you wrote that.
grungy49 Jul 20, 2006	It is a good teaser, but it has multiple solutions. I looked at it again and I could see how you said that it was 1 in 3. By saying that at least one of the coins came up heads, does that mean there is a 1 in 2 chance that it also came up both heads? If this applies, then I think it would actually be a 1 in 1.5 chance of getting both heads. I'm not too sure about this strategy, though.
gromney Aug 07, 2006	There really is not controversy here. This one is easy to do on paper because there are not many possible out comes. So lets just go through the answer with a little more detail. There are four possibilities when flipping both coins. They are: Coin 1 \| Coin 2 Heads \| Heads Heads \| Tails Tails \| Heads Tails \| Tails Since at least one of the coins is heads, Tails\|Tails can be eliminated leaving you with three possible coin flips that fit this scenario: Coin 1 \| Coin 2 Heads \| Heads Heads \| Tails Tails \| Heads Now we need to find out how many of the three above possible coin flips where "the other coin is also heads." In other words, where both coins come up heads. When we look at the possible coin flips, there is only one where both are heads. So the answer is one out of three, or 1/3. This may seem counterintuitive, but it is the correct answer nonetheless.
Dedrik Aug 09, 2006	Okay here is a nice sample space, of tosses for a pair of coins, with 78 entrants TH HT TH HH HT HH HH TT TH HT TT TT TT TH TH HH TH TT HH TH TT TH TH HH HT HH HH TT HH HT HT TT HT TH TH HH TT TT TT HH TT HH TT TH TH HH TH HT HT HH TH HT HT TH TT TT TH HH TT HT TH HH HH TH TH TT TT HT HH TH HT TT HT TH TH HH HT HH 19 of these are TT so they cannot be used. That leaves us with 59 Of those 59, 21 of them are double heads. 21/59 is almost 36% very close to 1/3 not so close to 1/2 If you don't trust me try it yourself.
grungy49 Aug 26, 2006	I don't see the difference between the combination HT and TH. It doesn't matter. This means the only other chances are HH or HT. I don't think it is 1/3. Plus this is a duplicate of #3230 (Dogs).
Winner4600 Aug 27, 2006	grungy, there are TWO coins. Say Coin A is Heads and Coin B is Tails. That is different from Coin A being Tails and Coin B being Heads.
ztodd Oct 21, 2006	I knew when I read this teaser that there would be controversy. Probablility teasers can do that sometimes. I knew it would be more fun to read everyone's comments than to actually solve it. Consider these different ways it could have been worded: A. Suppose I flip a coin and it came up heads. Then I flip another coint. What is the probability that this 2nd coin is also heads? B. Suppose I flip two coins without letting you see the outcome, and then I tell you that the first coin came up heads. What is the probability that the other coin is also heads? C. Suppose I flip two coins without letting you see the outcome, and I tell you that at least one of the coins came up heads. What is the probability that the other coin is also heads? D. Suppose I flip a dime and a penny without letting you see the outcome, and I tell you that the dime and the penny were not both tails. What is the probability that both the dime and the penny turned up heads? The answer to A and B is 1/2. The answer to C and D is 1/3. Take a look at which one was the actual wording. Does anyone still think that saying "the other coin is also heads" if different than saying "both coins are heads"? They amount to the same thing, right?
ztodd Oct 21, 2006	You could also replace "at least one of the coins" with "either the first or second coin" on C.
PatH Feb 06, 2007	For any of you still working on this one. The main thing to remember is that when you flip two coins, you are twice as likely to get a heads and a tales, than you are to get two of a kind. This is true because the heads and tales combination can be accomplished two ways, while heads/heads and tales/tales can only be accomplished one way. This is the FIRST part that has to be understood in solving this puzzle. Also, there is no excuse for not TRYING this at home, it takes only moments with a couple of coins to prove out this concept (flip at least 25 times). -Pat
(user deleted) Jun 21, 2007	Come on now. The first coin is already chosen, there are only two sides to the second coin. It is without a doubt 50%
jon_joy_1999 Jun 23, 2007	at first, I thought the answer was 1/4, and reading the answer, I thought the answer was wrong, but having it argued in here, I am convinced that not only is the answer correct, but the question is phrased properly. the people who believe it is 1/2 are reading too much into the question. I'll explain it one event at a time, listing ALL possible results from each event EVENT 1: Suppose I flip two coins without letting you see the outcome, results: HH, HT, TH, TT EVENT 2: and I tell you that at least one of the coins came up heads. results: HH, HT, TH alternately stated: for a fact, one coin is heads, the other coin is unknown at this time. it is not stated that the coin in this event is the first coin, or the second coin. this is the inverse-parallel of the "I have two coins that equal $0.30 in my pocket. one of them is not a nickel, what are the two coins?" question EVENT 3: What is the probability that the other coin is also heads? results: HH. this is the only answer that satisfies all three events. I originally thought it was 1/4 because I had included TT as a possible result, but it is explicitly excluded in event 2
masquerademe235 Aug 28, 2007	I also don't agree with this teaser because although 1 coin has flipped heads, the other coin doesn't depend on the first. THE SECOND COIN IS BASED ON AN INDEPENDENT PROBABILITY. If it is 50% to get heads, then the answer is 1/2.
srost24 Feb 06, 2008	This question is worded wrong. The results of the first coin is TOTALLY irrelevant. If P=O/E, then P= 1 possible outcome out of 2 expected (it can only be Heads or Tails)...so 1/2 or 50% is the correct answer. Even if the first coin was tails, the answer on the second coin would still be 1/2.
tsimkin Feb 06, 2008	srost24 -- The question is worded correctly, and the answer given is the right answer. Think instead of the case where I have 10,000 fair coins, and I flip them all, and I tell you NOT that the first 9,999 came up heads, but that I have "at least 9,999 heads". What is the probability that they are all heads? By Bayes Theorem, it is 1/10,001. This is that same problem, only severely reduced.
Gale Nov 24, 2008	I'm sorry but you got this one wrong. Your could fix it by changing it just a little. You could say, "I toss the two coins and then you ask me, is at least one a head, to which I truthfully reply, yes". Now the probability for two heads is 1/3 the way you state it 1/2 is the better answer.
Gale Nov 26, 2008	There is a similar puzzle about a nearsighted gambler. He tosses the dice to the other end of the crap table but he can't see that far. So he asks, is there at least one 5 down there? Siomeone answeres, "yes". The probability for two 5's is now 1/11 But, if he asked instead, "What do do see down there"? And someone answered, "I see at least one 5". Then the probability for two 5's is 1/6
mrmanuke Nov 28, 2008	I hate to be just one more drop in the 1/2 bucket, but there is a severe flaw in the 1/3 logic. The users who created a computer simulation must have built this flaw into their programs. The flaw is in assuming that the player of this game knows that the possibility of TT is 0%. That is, you assume the person who flips the coin has somehow rigged the game so that TT would not occur. Here is another way to explain this logic flaw. Say the coin flipping person decided to play this game with several different people. He played only one time with each person, and the players were not aware of his previous rounds with other players. Every round he plays this game, he announces that one of the coins is heads or that one is tails. He can't rig the game, so TT is a valid possibility. Therefore, he is sometimes forced to tell the player "at least one of the coins came up tails." By the "1/3" logic, each of the players should choose the answer opposite to what the coin flipper says. If he says "there's at least one heads" then the player should choose tails by eliminating TT from the four possibilities. If the flipper says "there's at least one tails" then vice-versa. Here is the interesting result. The "1/3" logic tells us that 2/3 of the time, the player will be safe with the opposite choice. In other words, 2/3 of the time, the flipper (by chance) flipped both coins opposite (a TH or HT), and only 1/3 of the time flipped both coins the same (a TT or HH). But we know that that can't be. Despite the fact that the flipper announces one of the coins, he can't change the fact that 1/2 of the time both coins will come up the same. Since we can't accept the outcome of this "1/3" logic, we have to look at the problem differently. Some users have stated that there are 4 equal possibilities, HH, TT, TH, and HT, and that saying "at least one is heads" eliminates only TT. However, it is correct to eliminate both TT and TH. The reason is that saying "one is heads" is the same as labeling the coins as "one" and "the other". In fact, the coin flipper may as well show you the coin that is the "heads" coin to clearly demonstrate that the two coin flips are independent events. Then you can label the coins "left and right" "one and two" or whatever. If you've labeled them, then I think everyone can agree that TT and TH must both be eliminated.
Gale Nov 28, 2008	M/duke got it just right. The proponents of the 1/3 answer assume the flipper of the coins simply ignores the instances when TT occurs. He simply skips right past those TT tosses and flips again and again until he in fact can announce that he has indeed tossed at least one head.
Gale Nov 28, 2008	Sorry I got the name wrong, it's mrmanuke
javaguru Dec 21, 2008	The problem with the wording is that we have to assume that what we've been told about the outcome is arbitrary. If we are always told there is at least on tails when there are any tails, then the second coin will always be heads. If what we are told always depends on the first coin flipped, then the answer is 1/2. If which coin we are given information about is chosen randomly, then the answer is 1/3. The problem doesn't specify how the information were given is chosen, so there isn't enough information to answer the question.
Gale Dec 22, 2008	Exactly! It's just like the puzzle where you meet a woman at the airport and learn through conversation that she is the mother of exactly two children. In the firsst instance, you might ask her if at least one of her children is a boy. Or, in the second instance, you might ask her to disclose the sex of one of her children, chosen at random, from the two. You learn she is the mother of at least one boy. Now the answer is 1/3 in the first instance and 1/2 in the second.
opqpop Jan 27, 2010	Good puzzle for someone who's never studied probability. Otherwise, this is a very fundamental example that anyone who studied probability should easily be able to see.
racoonieboy Jul 26, 2010	In my opinion, it's 1/2. It's worded poorly. It asks what the probability of the other coin getting a heads is. In that case, it doesn't matter what the first one was. Thus, making the answer 1/2. If it's not, someone please answer why not.
racoonieboy Jul 26, 2010	I'm sorry. I misunderstood the question. : )
mykulvee Sep 08, 2010	I toiled over this. After reading the comments and solutions, I finally will agree to 1/3. But it makes me ask the question. If flipping one coin twice and taking and disregarding all tt results, is the probability of hh still 1/3?
opqpop Sep 24, 2010	On revisit, I'd say this question is poorly phrased when it says "the other coin"
zembobo Oct 28, 2011	Mykulvee, yes it's still 1/3. If the first flip is T then the second flip has to be H to count as a trial (to keep tru to the scenario). BUT YOU STILL HAVE TO TRY B/C IF IT COMES OUT T FOR THE SECOND FLIP THEN YOU DON'T COUNT IT AS A TRIAL! A lot of people get caught up on that. If the first flip is H, then the second is allowed to be either H or T. There's your 3 possible outcomes. TH...HT,HH. Remember people that TT is not a possible outcome as per the scenerio description.
zembobo Oct 28, 2011	This teaser is worded perfectly. Write your program like this: 1: flip two coins 2: are they both tails?
zembobo Oct 28, 2011	This teaser is worded perfectly. Write your program like this: 1: flip two coins 2: are they both tails?
zembobo Oct 28, 2011	This teaser is worded perfectly. Write your program like this: 1: Flip 2 coins 2: Is at least one of them heads?
zembobo Oct 28, 2011	Sorry I don't know why it keeps adding the comment before I'm done. Anyway, 1: flip two coins 2: if both are tails then goto step 1 because our scenario gives information that does not allow for TT situations. 3: add 1 to counter "X" remark: we know at this point that one of them is H and this is therefore a valid trial so we add 1 to our valid trial counter "X" 4: if both are H then add 1 to counter "Y" remark: counter "Y" represents the number of HH outcomes 5: goto step 1 Now set this program to run 100 times. X should come out to be close to 75 (TT came up 25 times) and Y should come out to be close to 25. Now take Y/X. So there is a 1/3 chance that both coins are H when you know that one of them is H (the program knows that one of them is H if they are not both T). Simple and unbiased. Run this on your TI-85 or 83 or whatever you have before you comment. You'll see.
javaguru Jan 07, 2012	@Zem, please see my earlier comment. BTW, Monte Carlo is not needed to solve this.