First, let's find the overall probability of getting heads after selecting a coin and flipping it.
Let's define:
Event A: Selecting coin i with probability 2^-i.
Event B: Getting heads when flipping coin i with probability 3^-i.
Thus, the probability of getting heads overall can be expressed as the sum of the probabilities of getting heads for each coin weighted by the probability of selecting that coin:
P(Heads) = Σ [P(coin i) * P(Heads | coin i)] for i from 1 to ∞
P(Heads) = Σ [2^-i * 3^-i] for i from 1 to ∞
Using the geometric sum formula:
Σ α^i = α / (1 - α) when |α| < 1
In this case, α = 2/3, which is less than 1, so we can use the formula:
P(Heads) = (2/3) / (1 - (2/3)) = (2/3) / (1/3) = 2
Therefore, the probability that the result is Heads is 2.
