Really confusing seemingly simple probability question
Q. You toss 2 coins at the same time. They land on the floor. You don't look at the coins. Your friend tells you that at least one coin is heads. What is the probability of the other coin being tails?
A1. [b]1/2 chance, or 50%[/b]
The first answer is the common answer someone would say when asked this question. Because the outcome of one coin doesn't affect the other, the probability of the other coin being tails is simply 50%.
A2. [b]2/3 chance, or 66.7%[/b]
When 2 coins are tossed, there are 3 possible outcomes:
H H - 1/4
H T - 1/2
T T - 1/4
Because it is given that at least one of the coin is heads, we can cross out the 'T T' outcome, so we are left with
H H - 1/3
H T - 2/3
Which makes it so that the probability of the second coin being tails is 66.7%, which contradicts the first answer.
Which is the right answer?
17 Comments • Newest first
@bloodIsShed That video asks a different question.
The video asks: "Given that one coin is heads, what is the probability that the other coin is also heads."
The question being asked in the OP's original post is: "Given that at least one coin is heads, what is the probability that the other coin is tails?"
Subtle difference, but different nonetheless.
The video asks a question that has three possible cases, as he explains graphically.
The OP is asking a question that only has two possible cases. Same or different.
If they're both heads, that's one case, if one is heads and the other is tails, that's a second case. Doesn't matter which is which.
There IS no third case in this situation because the question is only asking the probability of them being different, knowing one is heads.
One result out of two possible cases is 50%
This is why it's important for questions to be clearly and specifically worded, and why so many tests are bad.
It would be interesting to put the OP's question to the guy who made the video and see if he agrees there's a subtle difference in the questions.
[quote=BobR]No way. That's not what the question asked.
The original question was:
"Q. You toss 2 coins at the same time. They land on the floor. You don't look at the coins. Your friend tells you that at least one coin is heads.
[b]What is the probability of the [i]other[/i] coin being tails?[/b]"
The known coin has no effect on the probability of the other coin being tails.
Since you're addressing a coin whose condition is unknown, the probability that coin will be tails is 50%.
Had the question asked what is the probability that "one" coin is tails, the answer would be different.
As stated however, the probability of the other coin being tails is 50%[/quote]
No, no, no. You're thinking in the event that the two coins are tossed one after another. This is not the case. The problem is that we don't know which one (or both) is heads.
Your post implies that we know which one is heads. This is NOT the case. We have to consider the event as a set of two elements, one for each of the two coins. The total event space is {(coin A = heads, coin B = heads), (coin A = heads, coin B = tails), (coin A = tails, coin B = heads), (coin A = tails, coin B = tails)}
since we know that at least 1 of the coins is heads, the event (coin A = tails, coin B = tails) cannot happen. Since we're interested in the case that the other coin (we don't know which one) is tails, we take the number of events we're interested in, and divide it by the total number of event spaces. In this case, that's 2/3.
similar problem [url=http://www.youtube.com/watch?v=4PwnvqGEHoU] skip at 3:15[/url]
[quote=BobR]The probability of the second coin being tails is 50%.
However the first coin lands doesn't have any effect on how the second coin is going to land.
Your original question is about ONE coin.
50%
That is a DIFFERENT question. Now you're asking what is the probability of one out of 4 possible results.
25% that they're both heads.[/quote]
He asks the questions in a pretty vague manner and his followup post to mine seems to be asking something else, so... Making confusing things more confusing.
[quote=bloodIsShed]False. The coins are tossed at the same time, and you don't know which one (or both) is heads. This is a simple conditional probability problem.
The probability that one of the coins is tails given that the other coin is heads is 2/3.[/quote]
No way. That's not what the question asked.
The original question was:
"Q. You toss 2 coins at the same time. They land on the floor. You don't look at the coins. Your friend tells you that at least one coin is heads.
[b]What is the probability of the [i]other[/i] coin being tails?[/b]"
The known coin has no effect on the probability of the other coin being tails.
Since you're addressing a coin whose condition is unknown, the probability that coin will be tails is 50%.
Had the question asked what is the probability that "one" coin is tails, the answer would be different.
As stated however, the probability of the other coin being tails is 50%
[quote=BobR]The probability of the second coin being tails is 50%.
However the first coin lands doesn't have any effect on how the second coin is going to land.
Your original question is about ONE coin.
50%
[quote=bob32]You know that one of them is heads. What is the probability that both of them is heads?[/b]
That is a DIFFERENT question. Now you're asking what is the probability of one out of 4 possible results.
25% that they're both heads.[/quote]
Even if this is slightly different to the original question, the answer isn't 25%. It can either be 50% or 33.3% depending on how you look at it. It would only be 25% if it was simply just tossing 2 coins.
[quote=BobR]The probability of the second coin being tails is 50%.
However the first coin lands doesn't have any effect on how the second coin is going to land.
Your original question is about ONE coin.
50%
[/quote]
False. The coins are tossed at the same time, and you don't know which one (or both) is heads. This is a simple conditional probability problem.
The probability that one of the coins is tails given that the other coin is heads is 2/3.
The probability of the second coin being tails is 50%.
However the first coin lands doesn't have any effect on how the second coin is going to land.
Your original question is about ONE coin.
50%
[quote=bob32]You know that one of them is heads. What is the probability that both of them is heads?[/quote]
That is a DIFFERENT question. Now you're asking what is the probability of one out of 4 possible results.
25% that they're both heads.
[quote=banana162825]hmm i guess it all depends on perspective of the problem and what you want to know from this answer i guess.
probability is relevant to cheese
btw why do you want to know?
if event emcompasses possibilities before throwing - certain number
if event emcompasses possibilities after landing - certain number
if event emcompasses possibilities of the result of certain event within event- certain number[/quote]
I think this is the worst answer out of all of the answers so far lol.
It's 2/3 since your outcomes are HH,HT and TH given the friends input. It would only be 50% if they were tossed one at a time or if you were given heads first and you only had to worry about the 2nd coin.
Also I wouldn't double up TH and HT as the same thing. In probability having one come first as opposed to the second can mean completely different things. In a simple case like this its fine as you're given the case that your friend says at least one but in other problems the 1/4 from HT and 1/4 from TH won't be the same.
It's 50% because you're just asking about that one coin so the other coin is completely irrelevant.
hmm i guess it all depends on perspective of the problem and what you want to know from this answer i guess.
probability is relevant to cheese
btw why do you want to know?
if event emcompasses possibilities before throwing - certain number
if event emcompasses possibilities after landing - certain number
if event emcompasses possibilities of the result of certain event within event- certain number
Probability is annoying, humans suck at it. If I know at least one of them is heads then the possibilities are:
H H
H T
T H
Because I don't know if the heads is necessarily on the left or the right hand side, am I right? I think you need to edit your question, it's not clear enough.
But let me tell you. Most confusing probability question is the goat question. It's on Wikipedia.
[quote=Darkwizzie]Question felt a little vague to me. (It's 5 in the morning.)
'Q. You toss 2 coins at the same time. They land on the floor. You don't look at the coins. Your friend tells you that at least one coin is heads. What is the probability of the other coin being tails?'
What does it mean when you ask 'what is the probability of the other coin being tails' what coin is the 'other' coin? The friend doesn't point to one coin and say that is heads or tails, he said at least one.[/quote]
You know that one of them is heads. What is the probability that both of them is heads?
Question felt a little vague to me. (It's 5 in the morning.)
'Q. You toss 2 coins at the same time. They land on the floor. You don't look at the coins. Your friend tells you that at least one coin is heads. What is the probability of the other coin being tails?'
What does it mean when you ask 'what is the probability of the other coin being tails' what coin is the 'other' coin? The friend doesn't point to one coin and say that is heads or tails, he said at least one.
[quote=banana162825]wrong because there are 4 combinations if you assume the "other" coin can be any one of the two
HH
HT
TH
TT
so it would be 2/4 or 1/2 i win[/quote]
I don't really get what you're trying to say. I've already taken into account that there is both a 'HT' and a 'TH' by making 'HT' 1/2 instead of 1/4.
wrong because there are 4 combinations if you assume the "other" coin can be any one of the two
HH
HT
TH
TT
so it would be 2/4 or 1/2 i win
2/3
Hi There ... Not extending post at all