Leejc is absolutely right He did a better job expressing what I tried to say back in June 2007 But while I agree with his final conclusion it isn t quite the right argument to use that we must assume the worst to get it So I thought I would try once again to explain how the different wordings of this problem that seem to say the same thing can lead to different solutions The ultimate cause of confusion is the math specific definition of the term event When a random experiment produces a result that result is not called an event it is an occurrence Event is defined to be the subset of all possible occurrences that have some defining common property So if I flip a coin a hundred times there are only two possible events heads and tails There will most likely be about fifty occurrences of each event and each occurrence will fit into only one event If I draw a card from a standard deck of playing cards there are lots of possible events and or occurrences Some of them overlap They include black card spade honor card TJQKA Ace and Ace of Spades The occurrence black card can be in the event spade or club as well Note that an occurrence can fit into other events but not vice versa An occurrence of spade is also an occurrence of black card but the event spade is most definitely not the event black card Since determining probabilities involves counting the number of elements in events the difference can be crucial when less specific observations are applied There is a famous problem that isolates this difference between an occurrence and an event It is called Bertrand s Box Paradox Suppose you have three indistinguishable boxes each with two indistinguishable drawers Inside each drawer is a coin Box 1 has two gold coins Box 2 has two bronze coins I can t resist the temptation to use the initials B and G and Box 3 has one of each You pick a box at random and open a drawer at random If the coin is gold what are the odds that the coin in the box s other drawer is also gold i e that it is Box 1 The first issue here common to story problems is that this problem describes an occurrence not an event We must extrapolate what the possible outcomes of the actions are to define the event the set of possible occurrences that defines the conditional probability The incorrect approach to Bertrand s Box Paradox is to say that you narrowed the identity of the box to two possibilities Box 1 and Box 3 That the event matches the observation the box has at least one gold coin This is wrong because there are other ways to describe occurrences in that event If you pick box 3 then no matter which drawer you open the occurrence is in the event the box has at least one gold coin The correct approach is to define the event as the set of all possible occurrences where you open a drawer with a gold coin Every possible occurrence that is consistent with the problem description is in this event remember it s a set and every possible element of this event is consistent with the problem description Both of those conditions have to be true for the event set we extrapolate as the condition This set has three elements not two In two of those three possible occurrences the box s other coin is gold The correct answer is 2 3 I have seen a variation of Bertrand s Box Paradox used to represent the Boy or Girl Paradox You are the organizer of a neighborhood swim meet During the meet you get a phone call from Mr Smith who is delayed and can t pick up his children after the meet He says Tell my two kids to go home with a neighbor Their names are Sarah and and the phone connection goes dead There are four families of two children in the meet and they conveniently include the four possibilities BB BG GB and GG What are the odds that Mr Smith has two daughters This occurrence can be described as a two child family includes at least one girl But not the event you need to use as the condition Mr Smith could have mentioned a boy s name first and if that boy had a sister that occurrence would have been an element of the set a two child family includes at least one girl So that event fails one of the two criteria You get the answer 1 3 by erroneously using this event The correct event is just that Mr Smith mentioned that he had a daughter and compares to you opened a drawer with a gold coin in the Box Paradox You can count four possible occurrences of that event In two of those four representing a single family Sarah has a sister The correct answer is 2 4 or 1 2 If a conditional probability question describes an occurrence and not an event care needs to be taken whe