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)

(In reply to re(2): I think the solution is WRONG by Cory Taylor)

While Lewis's reason is oversimplified, the explanation offered by Cory assumes that the person making the announcement is going through this procedure, that is, taking any presence of a tail as an opportunity for announcing that at least one is a tail.

There may be equally many people eager to announce, and taking any opportunity to announce a head. In the absence of any evidence to the contrary, why are we to assume that preferential tail announcers even outnumber preferential heads announcers. We could be in the presence of a preferential heads announcer forced by two tails to report at least one is a tail.

Given eight occasions:
HH by pref head announcer; announces at least 1 heads
HT by pref head announcer; announces at least 1 heads
TH by pref head announcer; announces at least 1 heads
TT by pref head announcer; announces at least 1 tail
HH by pref tail announcer; announces at least 1 heads
HT by pref tail announcer; announces at least 1 tail
TH by pref tail announcer; announces at least 1 tail
TT by pref tail announcer; announces at least 1 tail

On the four occasions when the announcer calls at least one tail, in two instances the other coin is a heads and in two instances the other coin is a tail.

Why do you assume (as does the official solution) that any announcer will preferentially report "at least one tail" over reporting "at least one head"?

By the way, it doesn't have to work out based on preferential announcing. It could be he chooses which to announce based on the first flip, or the coin that lands nearest him, or the one that's a quarter rather than a nickel. However he makes a decision as to which one to report, it still comes out to probability 1/2, except if (and this is not stated in the problem) the narrator who hasn't seen the coin, asks the announcer "Was at least one a tail?" thereby turning the announcer into a preferential tail reporter. But it nowhere in the problem states that this is what prompted the announcement.

Posted by Charlie on 2003-05-31 08:50:14