Probability

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. 👍
  2. 👎
  3. 👁
  1. 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

    1. 👍
    2. 👎
  2. Other answers please.

    1. 👍
    2. 👎
  3. 3a) 20/27
    3b) 2/3
    5) 1/9

    1. 👍
    2. 👎
  4. thanks

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    A biased coin lands heads with probabilty 2/3. The coin is tossed 3 times a) Given that there was at least one head in the three tosses, what is the probability that there were at least two heads? b) use your answer in a) to find

  2. Math

    Four coins fall onto the floor. Find the probability that (a) exactly three coins land heads up (b) all coins land tails up (c) two or more coins land heads up (d) no more than two coins land tails up (e) at least one coin lands

  3. math - probability

    a) What is the probability of obtaining 2 Heads when a coin is tossed twice? b) What is the probability of obtaining 1 Head when a coin is tossed twice? Keep in mind, the coins are not tossed simultaneously.

  4. Math: Probablity

    A penny, a nickel, and a dime are flipped at the same time. Each coin can come out either heads (H) or tails (T). a. What name is given to the act of flipping the coins? b. There are eight elements in the sample space (i.e.,

  1. probability

    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

  2. Math

    "The probability of getting heads on a biased coin is 1/3. Sammy tosses the coin 3 times. Find the probability of getting two heads and one tail". I thought that all you have to do is: (1/3)(1/3)(2/3) It makes sense to me, but

  3. Algebra Probability

    A coin is tossed twice. What is the probability of tossing heads, and then tails, given that the coin has already shown heads in the first toss?

  4. Math

    10. A coin is loaded so that the probability of heads is 0.55 and the probability of tails is 0.45, Suppose the coin is tossed twice and the results of tosses are independent. a.What is the probability of obtaining exactly two

  1. probability

    You flip a fair coin (i.e., the probability of obtaining Heads is 1/2) three times. Assume that all sequences of coin flip results, of length 3, are equally likely. Determine the probability of each of the following events. {HHH}:

  2. Probability

    We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. We assume that conditioned on Q=q, all coin tosses are independent. Let

  3. Math

    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

  4. math answer check

    Three coins are tossed, and the number of heads observed is recorded. (Give your answers as fractions.) (a) Find the probability for 0 heads. Incorrect: Your answer is incorrect. . Your answer cannot be understood or graded. More

You can view more similar questions or ask a new question.