Prize Probabilities
In “A Random Walk through Middle Land” [Skeptic], Michael Shermer presents the following scenario: you are a contestant on Let’s Make a Deal and are shown three doors. Behind one is a new car; the others hide goats. You choose a door, and host Monty Hall reveals a goat behind a different door. Shermer then posits that you have a two-thirds chance of winning by switching your choice because there are only three possible door configurations (good, bad, bad; bad, good, bad; and bad, bad, good), and with the latter two you win by switching. But he fails to recognize that the second configuration has been taken off the table. You have a 50 percent chance.
Andrew Howard
Los Angeles
SHERMER REPLIES: In nearly 100 months of writing the Skeptic column, I have never received so many letters as I did disagreeing with my description of the so-called Monty Hall Problem. James Madison University mathematics professor Jason Rosenhouse, who has written an entire book on the subject—The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser (Oxford University Press, 2009)—explained to me that you double your chances of winning by switching doors when three conditions are met: 1) Monty never opens the door you chose initially; 2) Monty always opens a door concealing a goat; 3) When your initial choice was correct, Monty chooses a door at random. “Switching turns a loss into a win and a win into a loss,” Rosenhouse says. “Since my first choice is wrong two thirds of the time, I will win that often by switching.”
At the beginning you have a one-third chance of picking the car and a two-thirds chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens one third of the time, and switching doors is good if you initially chose a goat, which happens two thirds of the time. Thus, the probability of winning by switching is two thirds. Analogously, if there are 10 doors, initially you have a one-tenth chance of picking the car and a nine-tenths chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens one tenth of the time. So the probability of winning by switching is nine tenths—assuming that Monty has shown you eight other doors with goats.
Still not convinced? Google “Monty Hall Problem simulation” and try the various computer simulations. You will see that you double your actual wins by switching doors. One of my skeptical correspondents ran his own simulation more than 10,000 trials, concluding that “switching doors yields a two-thirds success rate while running without switching doors yields a one-third success rate.” (Go to http://tinyurl.com/bu9jl for the simulation.)
Read an expanded version of Shermer’s reply here.
Note: This article was originally printed with the title, "Letters to the Editors".
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6 Comments
Add CommentThere is some evidence to suggest that pregnant mothers may influence their embryonic child's intelligence in some way. Yet no one has proved that intelligence can be passed from fathers to children through sperm.
Reply | Report Abuse | Link to thisIs inherited intelligence only aquired from mothers?
Luke Conlin,
Reply | Report Abuse | Link to thisAre you suggesting that science starts lying so that smart pople are not recognized? There are thousands of people way smarter than me...Thank goodness they can do things I can't. Why is it a problem is for some people that we are all different. What a bore...if we were all the same.
eco-steve,
Reply | Report Abuse | Link to thisIt's unlikely because half of the genes are from each parents. The rest are trashed for good. ..We only use 50% of our genes each time and the rest are gone for good. Read up on it.
Your mother was wrong. Saying something three times does not make it come true.
Reply | Report Abuse | Link to thisGee... your sharp froynlaven:
Reply | Report Abuse | Link to thisYou spotted my flaky bouncing mouse. Now the question is ...can you spot what's flaky with the ideal that intelligence is not inherited. I. E. The man is making a case that there's no genetic influences on intelligence. If he didn't have an obvious liberal bent that's fuzzed up his brain, I would bet Robert thinks that all women are just a bunch of "spare ribs" left over from the Garden of Eden and they all have identical I.Q.'s.
Whatsup
I'm not saying everybody's just as smart as everyone else...I'm saying that (1) "smartness" is really not well defined, and (2) IQ tests don't measure "smartness" whatever it is, and (3) nurture may be way more important than nature in this case...which is of course surprising but that's what the scientific evidence seems to be saying!
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