The Problem of the Two Boys

A family has two children, and you know that at least one of them is a boy. What is the probability that both are boys? There are four possibilities altogether (boy-boy, boy-girl, girl-boy, and girl-girl), and we can eliminate the last, so it would seem that the answer is 1/3.

But now suppose you visit a family that you know has two children, and that a boy comes into the room. What is the probability that both children are boys? Of the two children, you know that this one is a boy, and there is a probability of 1/2 that the other is a boy. So it seems that there is a probability of 1/2 that both are boys.

How can this be? We seem to have the same amount of information in both cases. Why does it lead us to two different conclusions?

Din Minimum

In 1958, acoustician William MacLean of the Polytechnic Institute of Brooklyn answered a perennial question: How many guests can attend a cocktail party before it becomes too noisy for conversation? He declared that the answer, for a given room, is

cocktail party noise


N0 = the critical number of guests above which each speaker will try overcome the background noise by raising his voice
K = the average number of guests in each conversational group
a = the average sound absorption coefficient of the room
V = the room’s volume
h = a properly weighted mean free path of a ray of sound
d0 = the conventional minimum distance between speakers
Sm = the minimum signal-to-noise ratio for the listeners

When the critical guest N0 arrives, each speaker is forced to increase his acoustic power in small increments (“I really don’t know what she sees in him.” — “Beg your pardon?” — “I say, I REALLY DON’T KNOW WHY SHE GOES OUT WITH HIM”) until each group is forced to huddle uncomfortably close in order to continue the conversation.

“We see therefore that, once the critical number of guests is exceeded, the party suddenly becomes a loud one,” MacLean concluded, somewhat sadly. “The power of each talker rises exponentially to a practical maximum, after which each reduces his or her talking distance below the conventional distance and then maintains, servo fashion, just the proximity, tête à tête, required to attain a workable signal-to-noise ratio. Thanks to this phenomenon the party, although a loud one, can still be confined within one apartment.”

(William R. MacLean, “On the Acoustics of Cocktail Parties,” Journal of the Acoustical Society of America, January 1959, 79-80.)

Vanishing Act
Image: Wikimedia Commons

In 1913 mathematician P.E.B. Jourdain proposed a familiar paradox:

On one side of a blank strip of paper, write The statement on the other side of this paper is true.

On the other side, write The statement on the other side of this paper is false.

“The paradox in this form is quite vulnerable to an absolute refutation,” wrote Valdis Augstkalns in a 1970 letter to The Listener. “One takes the paper, gives it a half twist, and joins the ends to form a Möbius strip. The serious and philosophically legitimate question is transformed to ‘Eminent members of the panel, which is the other side of the paper?'”


  • A pound of dimes has the same value as a pound of quarters.
  • The French word hétérogénéité has five accents.
  • 32768 = (3 – 2 + 7)6 / 8
  • Can you deceive yourself deliberately?
  • “My country is the world, and my religion is to do good.” — Thomas Paine

In 2000, Guatemalan police asked Christmas revelers not to fire pistols into the air. “Lots of people die when bullets fall on their heads,” National Civilian Police spokesman Faustino Sanchez told Reuters. He said that five to ten Guatemalans are killed or injured each Christmas by falling bullets.