I toss two coins and look at the outcome. I then tell you that at least one of the coins is showing up as "tails". What is the chance that the other one is showing "tails" as well?

(from techInterview.org)

It’s 1/3
Or maybe 50-50
Like ‘Charlie says’ (British ad-joke) if you ask the flipper ‘is one of them tails?’ and he responds in the affirmative then you’ve ruled out the HH option and most likely the other coin shows heads.
The way the question is posed implies either
i) (How most have interpreted). It’s a one-time deal. The flipper has casually flipped two coins and one of them is a tail. That out of the way, ‘what is the probability the other is a tails?’
ii) The flipper only reports ‘at least one Tail’ (outcome HH means no report)
iii) The flipper only reports ‘at least one Tail’ (unless the outcome is HH in which case he reports ‘at least one Head’)
iv) The flipper randomly reports ‘at least one Head/Tail’ (ruling out the opposite double HH or TT)
v) The flipper can choose when and what to report

For case;
i) This could have happened in one of three ways, all equally likely. Probability of the other being a Tail is only one of those three options (prob(T) = 1/3)
ii) Like above, the other coin is clearly twice as likely to be Heads for the reason described in most of the points here (prob(T)=1/3)
iii) A report of Heads makes it a certainty the other is a Head (prob(T) =0). The other ¾ of the time (when he reports Tails) the other coin is twice as likely to be a Head (prob(T) = 1/2 [3/4*2/3])
iv) Half the time the coins will be different. Half of these times the flipper will report Heads (prob(T) = 2/3) and half of these times report Tails (prob(T) = 1/3)
Total probability = (1/4*2/3 + 1/4*1/3) = 1/4
The other half of the time the coins will show the same the flipper is forced to report the side they show and whether you choose the same or different,
For HH flipper forced to report one is Heads (prob(T) = 0*1/4 = 0)
For TT flipper forced to report one is Tails (prob(T) = 1*1/4 = 1/4)
Total probability the other coin is Tails = 1/4+1/4 = 1/2
v) The flipper knows the other coin is more likely to be Heads if ‘at least one is Tails’ and so only reports one is Tails if both are (prob(T) =1) in order to catch you out.

I agree the question’s more likely to be interpreted as case i), simply a casual coin-flipping exercise. My problem with it is it is very dull to think someone has flipped two coins – and with no thought for profit, interest or amusement reported one is a Tail and went about their business whilst you’ve pondered on the probability the other was a Tail. If this is the case then prob(T) is certainly a 1/3 but where there is human input there is motive and this is undoubtably a zero sum ‘game’ . Cases i) and ii) are biased in favour of the ‘player’ – iii) and iv) are fair and v) seems like a game of wits - but boils down to a fair game if you as a guesser simply pick H or T at random yourself.
I find it difficult to believe it’s either of the first two cases.
In a game environment, because of the human input, I feel, on a one-off trial, the flipper has reported ‘at least one Tail’ only when both are (knowing the player will believe the other to be Heads as twice as likely).
Of course as a guesser, you are aware of this.
Of course as a flipper, you are aware the guesser is aware
Of course as a guesser, you are aware the flipper is aware, the guesser is aware…………..

Posted by Lee on 2003-12-03 23:50:28