A few teasers

1. There is a hypothetical creature X. The probability that X splits into two offsprings is p and that it dies without producing any offspring is 1-p. What is the probability that the family tree of X will go on forever?

2. If the sum of a set of positive integers is 2000. What can be the maximum possible product of the constituent integers?

3. A and B decide to meet up in front of city centre on 25^th April. They decide that each should arrive between 5PM and 6PM and whoever arrives first, will wait for 15 minutes for the other to turn up. If the other doesn’t turn up within those 15 minutes, he will leave. What is the probability that they successfully meet each other?

4. There are 124 prisoners. The jailor one day tells all of them, that he has a room which has a light bulb connected to a switch. The bulb is initially off and the room is not visible from any of the cells. He also said that next day onwards, the prisoners will be put in solitary confinement cells and there will be no communication of any sort possible between them. The jailor will each day, pick one prisoner at random and send him to that room with the bulb. He has the option of switching it on/off if he wishes. He also has the option of confirming to the jailor whether all 124 prisoners have been there in the room at least once. If he is correct, all of them would be set free. In case he is wrong with his assertion, all would be shot. The prisoners are then allowed to have a meeting to decide on a strategy before they are sent to the solitary confinement cells. What can be a strategy that the prisoners can decide on, which would guarantee their freedom?

5. There are three identical boxes with two coins in each. One has two gold coins, another has two silver coins and the third one has one gold and a silver coin. You pick one box at random and without peeping in, take a coin out. It turns out to be a gold coin. What is the probability that the other coin in the same box will also be a gold coin?