Assume that the prize is placed in a box at random, and that Fred makes his choice at random. Assume also that the host knows where the prize is, and that he would not have revealed its location by accident.
Calculate the conditional probability that box A contains the prize, given what Fred learns from the host. Explain carefully why Fred might act as he does.
I have two coins in my pocket. One of them is fair (probability 1/2 of landing heads on any toss) and the other is weighted to land heads with probability 2/3. I choose a coin at random and toss it twice. Write H1 for the event that the first toss lands heads and H2 for the event that the second toss lands heads.
The kingdom being aflicted with a surfeit of males, the ruler declares a new reproductive policy:
(a) no couple may have more than four children;
(b) after the birth of a male child, a couple may produce no more children.
Assume that each new birth has probability 0.5 of producing a male, regardless of previous reproductive history, and that all couples continue to produce children until either they have a son or they reach the maximum number of 4 children.
Write X for the number of female offspring in a family and Y for the number of male offspring.