To solve these problems, we need to apply conditional probability and basic probability principles. Let's go through each of the questions step by step:
1. To find the conditional probability that Bob received the first coin given that he observed k Heads out of the 3 tosses, we can use Bayes' Theorem:
P(First Coin | k Heads) = (P(k Heads | First Coin) * P(First Coin)) / P(k Heads)
P(k Heads | First Coin) is the probability of observing k Heads given that Bob received the first coin. Since the probability of Heads for the first coin is 1/4, this can be calculated as (1/4)^k * (3/4)^(3-k) * (3 choose k), which represents the probability of getting k Heads and (3-k) Tails in any order.
P(First Coin) is the probability that Bob received the first coin, which is given as p.
P(k Heads) is the probability of observing k Heads regardless of which coin Bob received. We can calculate this probability by considering all possible scenarios:
P(k Heads) = P(k Heads | First Coin) * P(First Coin) + P(k Heads | Second Coin) * P(Second Coin)
= (1/4)^k * (3/4)^(3-k) * (3 choose k) * p + (3/4)^k * (1/4)^(3-k) * (3 choose k) * (1-p)
= (3 choose k) * [(1/4)^k * (3/4)^(3-k) * p + (3/4)^k * (1/4)^(3-k) * (1-p)]
So, the conditional probability that Bob received the first coin is:
P(First Coin | k Heads) = [(1/4)^k * (3/4)^(3-k) * p] / [(3 choose k) * [(1/4)^k * (3/4)^(3-k) * p + (3/4)^k * (1/4)^(3-k) * (1-p)]]
= (3^3-k * p) / (3^3-k * p + 3^k * (1-p))
2. To minimize the probability of error, Bob should choose the threshold condition that minimizes the conditional probability of error. Based on the given choices, the correct threshold condition is:
k ≤ 3/2 + (1/2) * log3(p / (1-p))
This threshold condition determines under which conditions Bob should decide that he received the first coin.
3.
(a) For this part, assume that p = 3/4. Plug in the value of p into the threshold condition from part 2 to find the specific condition for minimizing the probability of error.
(b) If Bob tries to guess the coin without tossing it, he will use the same threshold condition as in part 2. The probability that Bob will guess the coin correctly using this decision rule can be calculated as 1 - P(Error), where P(Error) is the probability of error. To find P(Error), we need to consider the scenarios where Bob decides he received the first coin but actually received the second coin and vice versa. We can use the threshold condition from part 2 to determine these probabilities.
4. If p is small, Bob will always decide in favor of the second coin, ignoring the results of the three tosses. The range of such p's can be expressed as [0, t), where t represents the upper bound of p. To find t, we need to find the value of p for which the threshold condition from part 2 becomes k > 3/2. This represents the point where Bob switches from deciding in favor of the second coin to deciding in favor of the first coin based on the number of Heads observed in the three tosses.