Example 2.19 (The Monty Hall problem).
The problem is named for the
host of the television show Let’s Make A Deal in which contestants were often
placed in situations like the following: Three curtains are numbered 1, 2, and
3. Behind one curtain is a car; behind the other two curtains are donkeys. You

pick a curtain, say #1.
To build some suspense the host opens up one of the
two remaining curtains, say #3, to reveal a donkey. What is the probability you
will win given that there is a donkey behind #3?
Should you switch curtains
and pick #2 if you are given the chance?

2.2.
CONDITIONAL PROBABILITY
47
Although it took a number of steps to compute this answer, it is “obvious.”
When we picked one of the three doors initially we had probability 1/3 of picking
the car, and since the host can always open a door with a donkey the new
information does not change our chance of winning.
The paradox actually predates the game show in the following form. Three
prisoners, Al, Bob, and Charlie, are in a cell. At dawn two will be set free and
one will be hanged, but they do not know who will be chosen. The guard offers
to tell Al the name of one of the other two prisoners who will go free but Al
stops him, screaming, “No, don’t!
That would increase my chances of being
hanged to 1/2.” Criticize Al’s reasoning.
People vs. Collins.
In 1964 the purse of an elderly woman shopping in Los
Angeles was snatched by a young white female with a blond ponytail. The theif
fled on foot but soon after was seen getting into a yellow car driven by a black
man who had a mustache and beard. A police investigation subsequently turned
up a suspect who was blond, wore a ponytail, and lived with a black man who
had a mustache, beard, and a yellow car. None of the eyewitnesses were able to
identify the suspects so the police turned to probability
Characteristic
Probability
Yellow car
0.1
Man with mustache
0.25
Black man with beard
0.01
Woman with ponytail
0.1
Woman with blond hair
0.33
Interracial couple in car
0.01
Multiplying the probabilities as if they were independent gives an overall prob-
ability of 1
/
12
,
000
,
000. This impressed the jury who handed down a verdict
of second degree robbery. The Supreme Court of California later disagreed and