>>12415717(2/2)
However, we get 3/7 by assuming every single possible event follow a flat distribution.
That would mean the probability of knowing only one coin is twice the probability of knowing both coins.
What if the probability of knowing only one coin is EQUAL to the probability of knowing both coins? (aka flat distribution of the number of coins known)
Therefore both cases would share 50% of the total set of events.
1 known coin -> 8 arrangements.
2 known coins -> 4 arrangements
Tt * 1/16
Th * 1/16
Ht * 1/16
Hh * 1/16
tT * 1/16
tH * 1/16
hT * 1/16
hH * 1/16
TT * 2/16
TH * 2/16
HT * 2/16
HH * 2/16
At least one is heads:
Ht * 1/16
Hh * 1/16
tH * 1/16
hH * 1/16
TH * 2/16
HT * 2/16
HH * 2/16
10/16
Both coins are heads:
Hh * 1/16
hH * 1/16
HH * 2/16
4/16
Probability both coins are heads given one coin is heads: 4/16 / 10/16 = 2/5.
So assuming another proportion of 1 known coins vs 2 known coins, we get 2/5.
So what is is?
1/2?
3/7?
2/5?
The thing here is we don't know this proportion, we don't know the probability of knowing only one coin, or both.
No information is given, at all, and no assumption can be made.
You can't just say "at least one is heads" is the result of knowing only one coin. You don't know that.
You would have to consider all possibilities.
However, you don't know the probability of either number of known coins, and can't assume a flat distribution.
Therefore, there is no one single answer.
That would render the problem incomplete, with an infinite number of solutions.
That is following your line of logic that you need to know how the information was obtained.
Or you can just interpret this problem like every other textbook problem and let the information given by the predicate become the hypothesis which sends you on the path to one single answer.
Which is 1/3.
