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Sometimes, finding an answer to a probability question may be
challenging and create a controversy. To understand the problem
better and to determine the strategy for the solution, simulating
the situation can help. In this activity you will learn importance
of simulation in solution of probability questions.
Marilyn vos Savant, who is listed in Guinness Book of World
Records Hall of Fame for "Highest IQ", writes a
monthly column in the Sunday newspaper supplement Parade Magazine.
Her column, "Ask Marilyn", is devoted to games of skill,
puzzles, and mind-binding riddles. In one of the issues (Parade
Magazine, Feb., 17, 1991), vos Savant posed the following
question ( It is also known as Monty Hall problem and is named
for the host of the television show Let's Make A Deal which
was from 1963 to 1990):
Suppose you're on a game show, and you're given a choice of three
doors. Behind one door is a car; behind the others, goats. You
pick a door-say, #1-and the host, who knows what's behind the
doors, opens another door-say, #3-which has a goat. He then says
to you, "Do you want to pick door #2?" Is it your advantage
to switch your choice?
Marilyn answers: "Yes, you should switch. The first door
has a 1/3 chance of winning the car, but the second has a 2/3
chance of winning the car."
Needless to say, vos Savant's surprising answer led to thousands
of critical letters disagreeing with her. Many of the letters
were from Ph.D. mathematicians; some of the more interesting and
critical letters, that were printed in her next column, are condensed
here:
"Your logic is in error, and I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns." (Georgia State University)
"May I suggest you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?" (University of Florida)
"You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error you will have contributed constructively toward the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?" (Georgetown University)
"I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake." (Dickinson State University)
"You are the goat!" (Western State University)
"You're wrong, but look on the positive side. If all the
Ph.D.'s were wrong, the country would be in serious trouble."
(U.S. Army Research Institute)
The logic employed by those who disagree with vos Savant is as
follows: Once the host shows you door #3(a goat), only two doors
remain. The probability of the car being behind door #1(your door)
is 1/2; similarly, the probability is 1/2 for door #2. Therefore,
in the long run, it does not matter whether you switch to door
#2 or keep door #1. Approximately 50% of the time you will win
a car, and 50% of the time you will "win" a goat.
Who is correct, the Ph.D. mathematicians or Marilyn? What would
you do, switch or not switch?
a. First let us take a class field trip to the Monty Hall's
show. Click on the button to visit the Monty Hall's Homepage.
Play the game by using two strategies: (i) do not switch, (ii)
switch every time. For each strategy play the game at least 20
times and keep records of you winnings and losing.
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b. What is the probability of winning the car if you do not switch?
c. What is the probability of winning the car given that
your strategy is to switch?