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Conceptual Test: Probability

Can you find the fallacy in the following joke?

A statistic professor plans to travel to a conference by plane. When he passes the security check, they discover a bomb in his carry-on-baggage. Of course, he is hauled off immediately for interrogation.

"I don't understand it!" the interrogating officer exclaims. "You're an accomplished professional, a caring family man, a pillar of your parish - and now you want to destroy that all by blowing up an airplane!"

"Sorry", the professor interrupts him. "I had never intended to blow up the plane."

"So, for what reason else did you try to bring a bomb on board?!"

"Let me explain. Statistics shows that the probability of a bomb being on an airplane is \(10^{-3}\). That's quite high if you think about it - so high that I wouldn't have any peace of mind on a flight."

"And what does this have to do with you bringing a bomb on board of a plane?"

"You see, since the probability of one bomb being on my plane is \(10^{-3}\), the chance that there are two bombs is \(10^{-6}\). If I already bring one, the chance of another bomb being around is actually \(10^{-6}\), and I am much safer..."

Note by Agnishom Chattopadhyay
2 years, 7 months ago

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I'd be very worried about 1) air travel, if the probably of having a bomb on board is as high as \({ 10 }^{ -3 }\), and 2) quality of college education, if we have statistic professors bringing on board more bombs based on such deeply flawed thinking!

I should avoid sleeping in my bed tonight, because it's a statistical fact that most people die in their beds. Michael Mendrin · 2 years, 7 months ago

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@Michael Mendrin Indeed, given that there are roughly \(100,000\) flights per day around the globe, a probability of \(10^{-3}\) would mean that planes would be blowing up at a rate of \(4\) per hour, give or take. I think the actual odds would be closer to \(10^{-8}\). The professor is more likely to die falling out of bed than out of an exploding plane.

As for dying in bed, there are much worse ways to go, so best to take a chance and get a good night's rest. :) Brian Charlesworth · 2 years, 7 months ago

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@Brian Charlesworth Yes. Dying in my bed is more comfortable than dying on the floor. So, heck, I'll take my chances with my bed. Michael Mendrin · 2 years, 7 months ago

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The event \(A\) that the professor brings a bomb on board to make himself feel safer is independent of the event \(B\) that someone else brings a bomb on board for nefarious purposes, (assuming that the professor has no foreknowledge of the specific plans of any terrorists or terrorist organizations).

Thus the conditional probability \(P(B | A) = P(B) = 10^{-3}\), leaving the professor no safer than if he boards the plane without a bomb. Brian Charlesworth · 2 years, 7 months ago

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Hmmm, why did the professor not bring two bombs? It is clear that if the professor had brought all the bombs on the earth he would now be perfectly safe. So, is it unreasonable to assume that his safety grows greater with each bomb he brings? Bobbym None · 2 years, 7 months ago

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@Bobbym None Indeed. With all the bombs on Earth stashed in the plane it would be so heavy it could never take off. Then all the passengers would disembark, leaving the bombs behind and thus never have to worry again .... at least about being bombed from the sky. There might be some logistical problems, though. All the bombs on Earth taken together would probably fill the Grand Canyon; a sobering thought if ever there was one. :P Brian Charlesworth · 2 years, 7 months ago

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@Bobbym None Yeah! That is exactly right! Why not bring more bombs? Michael Mendrin · 2 years, 7 months ago

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Statistics are like a bikini. What they reveal is suggestive, but what they conceal is vital. - Quotation by Aaron Levenstein Rajen Kapur · 2 years, 7 months ago

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@Agnishom Chattopadhyay Did you make this,or did you get it from somewhere?If the latter ,please share the source. Adarsh Kumar · 2 years, 7 months ago

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@Adarsh Kumar It's a popular joke on the tubes, but I copied this exact one from statistics stackexchange. You'll have to wait till I can provide you the exact link Agnishom Chattopadhyay · 2 years, 7 months ago

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