Alice has two coins. The probability of Heads for the first coin is 1/4, and the probability of Heads for the second is 3/4. Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and sends it to Bob. The random selection used by Alice to pick the coin to send to Bob is such that the first coin has a probability p of being selected. Assume that 0<p<1. Bob tries to guess which of the two coins he received by tossing it 3 times in a row and observing the outcome. Assume that for any particular coin, all tosses of that coin are independent.
1. Given that Bob observed k Heads out of the 3 tosses (where k=0,1,2,3), what is the conditional probability that he received the first coin?
ans= 3^3−k*p/3^3−k*p+3^k*(1−p)
2. We define an error to have occurred if Bob decides that he received one coin from Alice, but he actually received the other coin. He decides that he received the first coin when the number of Heads, k, that he observes on the 3 tosses satisfies a certain condition. When one of the following conditions is used, Bob will minimize the probability of error. Choose the correct threshold condition.
ans= k≤3/2+1/2log3*p/1−p.
3. For this part, assume that p=3/4.
(a) What is the probability that Bob will guess the coin correctly using the decision rule from part 2?
ans= unanswered
(b) Suppose instead that Bob tries to guess which coin he received without tossing it. He still guesses the coin in order to minimize the probability of error. What is the probability that Bob will guess the coin correctly under this scenario?
ans= unanswered
4. Bob uses the decision rule of Part 2. If p is small, then Bob will always decide in favor of the second coin, ignoring the results of the three tosses. The range of such p's is [0,t). Find t.
t=
ans =unanswered
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.