The Car-and-Goats II
This is a continuation of the previous question... If the car is actually at Door I (probability 1/3), then when you switch you lose; but if it is at Door 2 or Door 3(probability 2/3) then the host's revelation of a goat shows you how to switch and win. Therefore the chance you win by switching is 2/3. Elegant. But in this argument, we are still considering the possibility that the car is behind Door 3; so the host cannot have already opened the door (much less to reveal a goat). In this game, you have to announce before a door has been opened whether you intend to switch.

The Car-and-Goats
A number of mathematicians were thrown into a tizzy by the following problem, which appeared in Marilyn vos Savant's column, "Ask Marilyn", in Parade(a Sunday newspaper supplement):

One of the three doors hides a car (all three equally likely) and other two hide goats. You choose Door 1. The host, who knows where the car is, then opens one of the other two doors to reveal a goat, and asks whether you wish to switch your choice. Say he opens Door 3, should you switch to Door 2?

Random Hats Three people are given hats. Each hat is either red or blue, chosen at random. Each person can see the other 2 hats, but not their own. They each must simultaneously either guess their own hat's color, or pass. No communication is allowed, although they can agree on a strategy ahead of time. What strategy will give them the best chances of at least one person guessing right, and nobody guessing wrong?

Probability While your back is turned, a friend places a penny, nickel, and dime on the table. He arranges them in a pattern of heads and tails provided that the three coins are not all heads or all tails.

Your object is to give instructions, without seeing the coins, that will cause all heads to be the same (all heads or all tails). For example, you may ask your friend to reverse the dime. He must then tell you whether you have succeeded in getting all the coins alike. If you have not, you again name a coin for him to turn. This procedure continues until he tells you that the three coins are the same.

Your probability of success on the first move is 1/3. If you adopt the best strategy, what is the probability of success in two moves are fewer? What is the smallest number of moves that guarantees success on or before the final move.


If you feel the above question easy, here is your question!
The situation is the same as before, except this time your intent is to make all the coins show heads. Any initial pattern except all heads is permitted. As before, you are told after each move whether or not you have succeeded. Assuming that you use the best strategy, what is the smallest number of moves that guarantees success? What is your probability of success in two moves or fewer, in three moves or fewer, and so on up to the final move at which the probability reaches 1 (certainly)?