Question

my Chosen from jar 2 inrk. Show that . the first apd ante all the probability ball is wife, i.e., Problem 23. We have twoj. We perform four successive ball exchane and at random a ball from cach jar and that at the end of the fou exchat successive ball exchanges, In each holl from each jar and move it to the other be fout exchanges all the balls will be in the in Protec The prisoner's dilemma. The rol S o unced, but their identity is kept secreto.. asking a friendly guard to tell him released, but hesitates based on the ailemma. The release of two out of three pr. identity is kept secret. One of the prisoners considers d to tell him who is the prisoner other than himself that will be ates based on the following rationale: at the prisoner's present state de the probability of being released is 2/3, but after he of being released will become 1/2, since there will be two prisoners mself) whose fate is unknown and exactly one of the two will be released. the probability of being released will become 1/2, since Whats wrong with this line of reasoning? Problem 25. ^/two-envelopes puzzle. You are handed two envelopes, and you Kurowthat each contains a positive integer dollar amount and that the two amounts are different. The values of these two amounts are modeled as constants that are unknown. Without knowing what the amounts are, you select at random one of the two envelopes, and after looking at the amount inside, you may switch envelopes if you wish. A friend he following strategy will increase above 1/2 your probability of ending up with the envelope with the larger amount: toss a coin repeatedly, let X be equal to 1/2 plus the number of tosses required to obtain heads for the first time, and switch if the amount in the envelope you selected is less than the value of X. Is your friend correct? Alzite Problem 26. The paradox of induction, Consider a statement whose truth is TITIKTUWI. twee many examples that are compatible with it, we are tempted to view the statut as more probable. Such reasoning is often referred to as induc- tive inference with a philosophical, rather than mathematical sense). Consider now the statement that all cows are white. An equivalent statement is that "everything that is not white is not a cow." We then observe several black crows. Our observations are clearly compatible with the statement, but do they make the hypothesis "all cows are white" more likely? To analyze such a situation, we consider a probabilistic model. Let us assume that there are two possible states of the world, which we model as complementary events: A: all cows are white, A : 50% of all cows are white. Let p be the prior probability P(A) that all cows are white: We make an observation of a cow or a crow, with probability q and 1 - 4, respectively, independent of whether

question 25

Answer 1

Solution:

Yes, My friend is right because both of the envolope has equal chances to choose.

P(To choose envolope with a amount) = 1/2

P(To choose envolope with b amount) = 1/2

Now if i choose envelope with a amount than probability is 1/2. Now i can choose the switch envelopes after watching it or not so its probability is also 1/2.

So the Probability of getting b amount = 1/2 * 1/2 = 1/4

So total probability of getting b amount envelope = 1/2(in one chance) + 1/4(In second chance)

Total probability = 0.75

SO my friend is right that probability increase from 1/2 to 3/4. or 0.5 to 0.75