Alice has two coins. The probability of Heads for the first coin is 1/3, and the probability of Heads for the second is 2/3. 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.
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?
123−k23−kp23−kp+2k(1−p)p23−k11+p1−p23−2k
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.
k≥32+12log2p1−pk≤32+12log2p1−pk≤12log2p1−pk≥12log2p1−pnone of the above
For this part, assume that p=2/3.
(a) What is the probability that Bob will guess the coin correctly using the decision rule from part 2?
- 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?
- unanswered
Suppose that we increase p. Then does the number of different values of k for which Bob decides that he received the first coin increase, decrease, or stay the same?
It increases or stays the same. - unanswered
Find the values of p for which Bob will never decide that he received the first coin, regardless of the outcome of the 3 tosses.
p is less than - unanswered
1. p*2^(3-k)
------------------------
p*2^(3-k) + (1-p)*2^k
2. k ≤ 3/2 + 1/2*log_2(p/(1-p))
4. it increases or stays the same
please, if you are reading these answers ...Dont be selfish and share the other answers to the whole problem set
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)).
thanks
Other answers please.
3a) 20/27
3b) 2/3
5) 1/9