We have an infinite collection of biased coins, indexed by the positive integers. Coin i has probability 2−i of being selected. A flip of coin i results in Heads with probability 3−i. We select a coin and flip it. What is the probability that the result is Heads? The geometric sum formula may be useful here: ∑i=1∞αi=α1−α, when |α|<1.

The probability that the result is Heads is

1/5.

P(heads) = P(heads|coin) * P(coin)
which is the multiplication of the two geometric sums
solving for 2-i * 3-i = 6-i or (1/6)^i. plugging this into the geometric sum formula we get 1/5

I wonder how you arrived to 1/5?

Geometric sum formula looks like this when using α=6^(-i)

6^(-i)/(1-6^(-i))

Because the first term of the geometric sequence is 1 => 1 + (-6/5) = -1/5 take the absolute value = 1/5

To find the probability that the result is Heads, we need to consider the probability of selecting each coin and the probability of getting Heads for each coin.

Let's break down the problem step by step:

1. Probability of selecting coin i: The probability of selecting coin i is given as 2^(-i). This follows a geometric series with a common ratio of 1/2. Using the geometric sum formula, we can express the probability of selecting any coin as follows:

P(selecting any coin) = ∑(i=1 to ∞) (2^(-i)) = 2^(1)/2 = 1/2.

2. Probability of getting Heads for coin i: The probability of getting Heads for coin i is 3^(-i). Again, this follows a geometric series with a common ratio of 1/3. Using the geometric sum formula, we can express the probability of getting Heads for any coin as follows:

P(Heads for any coin) = ∑(i=1 to ∞) (3^(-i)) = 3^(1)/3 = 1/3.

3. Probability that the result is Heads: To find the probability that the result is Heads, we need to consider the joint probability of selecting any coin and getting Heads for that coin. Since the two events are independent, we can multiply their probabilities:

P(Heads) = P(selecting any coin) * P(Heads for any coin) = (1/2) * (1/3) = 1/6.

Therefore, the probability that the result is Heads is 1/6.