To find the threshold condition for Bob to minimize the probability of error, we need to determine the number of heads, k, that he observes on the 3 tosses.
The formula given for the conditional probability that he received the first coin is:
P(First coin | k Heads) = p^k * (1-p)^(3-k) / [ p^k * (1-p)^(3-k) + (1-p)^k * p^(3-k) ]
In order to minimize the probability of error, we want to find the condition for k that maximizes P(First coin | k Heads).
To do this, we can take the derivative of P(First coin | k Heads) with respect to k and set it to zero. By simplifying the expression, we find that the threshold condition is:
k ≥ 3/2 + 1/2 * log2(p/(1-p))
Therefore, the correct threshold condition is k ≥ 3/2 + 1/2 * log2(p/(1-p)).