plzz...tell me the ans of this prob and the process how u solved it..?

Question

Of the subjects of a certain king, not all are truthful. In fact, if a subject is selected at random, the probability that he always tells the truth is 2/3. The probability that he never tells the truth is 1/3. The king of this country is trying to decide whom to marry. There are only two possible choices, Princess Anne and Princess Barbara. One day the king whispers to one of his subjects that his choice is Anne. This confidant hastily whispers to another person "The king has chosen ______". Which name he says depends, of course, on whether or not he is truthful. So it goes. Each person, when he hears the rumor, whispers either the name he hears or the other to someone who has not heard. Eventually, 12 people have heard the rumor. What is the probability that the 12th person has heard the truth?

Answer 1

Essentially, the 12th person has heard the truth if, after the whole chain, an EVEN number of people have "lied" (including 0)

Since it doesn't matter which people (as long as ANY 0,2,4,6,8 or 10 people lied)

So you need to sum the following probabilities:

P(0 lied) = (2/3)^11
P(2 lied) = (11choose2)(2/3)^9(1/3)^2
P(4 lied) = (11choose4)(2/3)^7(1/3)^4
... etc

for example 11choose2 = 11!/9!2! = 11*10/2 = 55

in general terms, MchooseN = M! / (N! * (M-N)!)

I'll let you plug the numbers into your calculator - good luck!

Answer 2

This is a binomial expansion problem: (P + Q)^n; where P = 2/3, Q = 1/3, and n = number of people hearing the rumor. Each term in the expansion represents the number of pathways of an event tree. Thus, for example, if n = 3 people hear the rumor, we have (P + Q)^3 = P^3 + 3P^2 Q + 3P Q^2 + Q^3 = 1.00

Of these four terms, P shows up in three. The fourth term (Q^3) cannot end up with the true (P = Anne) because there is no P in that pathway. The third term (3P Q^2) also cannot lead to P on the last person because the first exhange (the first P from the king) is the only P. (By the way, I presume the first P = 1.00, which means the king will in fact tell the truth. If the king is trying to mislead his kingdom, the P = 2/3 for the first rumor as well.)

Thus, only the first two terms can end with the last person hearing P, the truth. Let's look at them:

P^3: this one pathway clearly ends with the last person hearing the truth. This pathway looks like PPP; so the probability for it is 1(2/3)(2/3) = 4/9 (the first P = 1, the king will start the chain of events by telling truth).

3P^2 Q: there are three pathways here as indicated by the 3 coefficient. These, and their probabilities are...

PPQ = 1(2/3)(1/3) = 2/9
PQP = 1(1/3)(2/3) = 2/9
QPP = 0(2/3)(2/3) = 0

Thus, only the second pathway ends with the last person telling the truth. The last pathway, the person tells what he thinks is the truth, but it really isn't because the king lied to start with. But we discount that from ever happening; so its probability is zero. [If this is not your assumption, that P = 1 and Q = 0 for the king, then just change the P and Q values accordingly for the first rumor.]

Summing up, we have p(PPP) + p(PQP) = 4/9 + 2/9 = 6/9 which is the probability the last person will hear the truth.

Following this line of logic, you can work your problem with n = 12, the number of rumor mongers. Although n = 12 will give you considerably more terms in the expansion, many of the terms can be thrown out on inspection because they will not end with a P factor in last rumor. Clearly the PQ^11 and Q^12 terms will not end with P for the last rumor for example.

I'd recommend you look up Pascal's Triangle [See source.] for the coefficients of each term in (P + Q)^12. Then start with the P^12 term and work your way to the right to identify how many of the C(12,i) pathways of the coefficients are in fact ending with P in the n = 12 position of the pathway.

Answer 3

p that 1st person is telling truth 2/3.
p that 1st person is lying 1/3

p that 2nd person thinks it's Anne 2/3
p that 2nd person thinks it's Barb 1/3

p that 2nd person says it's Anne--tells the truth in the first instance and lies in the 2nd instance 2/3 * 2/3 + 1/3 * 1/3 or 4/9 +1/9 = 5/9.

p that 2nd person says its Barb--lies in the first instance and tells the truth in the 2nd instance 2/3 * 1/3 + 1/3 * 2/3 or 2/9 +2/9 = 4/9

p that third person thinks it's Anne 5/9
p that third person thinks it's Barb 4/9

p that third person says it's Anne --tells truth in the first instance and lies in the second instance 5/9 * 2/3 + 4/9 * 1/3 or 10/27 + 4/27 = 14/27

p that third person says it's barb--lies in the first instance tells the truth in the second 5/9 * 1/3 + 4/9 * 2/3 = 5/27 + 8/27= 13/27

P that 4th person hears Anne 14/27
p that 4th person hears Barb 13/27

p that 4th person says Anne--14/27 * 2/3 + 13/27 * 1/3= 28/81+ 13/81 = 41/81

p that 4th person says Barb --14/27 * 1/3 + 13/27 * 2/3= 14/81 + 26/81 = 40/81

You' should notice by now that with each person told that the probability approaches but never quite reaches 1/2.