# What is the logical/statistical switcheroo behind this joke? **URL:** \ **Category:** Factual Questions\ **Created:** [May 23, 2011, 10:20pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952 "2011-05-23T22:20:33Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [May 23, 2011, 10:20pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/1 "2011-05-23T22:20:33Z")
Ok. Guy is passing through inspection gate for a flight. It’s discovered he’s trying to smuggle a bomb aboard. When they interrogate him and ask him why he did it, he said “I’m terrified of a bombing on this flight. I figure, what are the odds of _two_ people bringing a bomb on board?” ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [May 23, 2011, 10:24pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/2 "2011-05-23T22:24:47Z")
It’s the independence of events. Once you are certain that a bomb is aboard, the chance of another becomes the chance of one bomb. The “chance of two bombs” is based on you not knowing whether or not there are ANY bombs on board. ---
**Author:** ![dracoi](https://avatars.discourse-cdn.com/v4/letter/d/90db22/32.png) [@dracoi](https://boards.straightdope.com/u/dracoi)\ **Post date:** [May 23, 2011, 11:35pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/3 "2011-05-23T23:35:41Z")
To put numbers to the previous explanation: Chance of one bomb, normally: 1% Chance of two independent bombs at once: 0.01% Chance of two bombs, given that we already know one is there: 1% ---
**Author:** ![Shagnasty](https://avatars.discourse-cdn.com/v4/letter/s/9dc877/32.png) [@Shagnasty](https://boards.straightdope.com/u/Shagnasty)\ **Post date:** [May 24, 2011, 12:04am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/4 "2011-05-24T00:04:42Z")
The answers given are basically correct but not completely. The chance of a passenger bomb aboard a random regular flight is higher than one you and your family are on because you took are taking up some of the seats on the plane (provided you and your family are not terrorists). That is just a nitpick however. Statistics of this type are fascinating. You think winning the lottery is rare? What about someone winning the lottery twice, There must be cheating going on right? Nope, when you run the statistics of all lottery winners, the odds are in favor of at least one of those prior winners winning again just by chance even if they aren’t lottery ticket junkies. Indeed, it has happened [multiple times.](http://www.abc15.com/dpp/news/national/Copy_of_Woman-wins-millions-from-Texas-lottery-for-4th-time_35170322-ews-SHNS-wmar1278098941198). Independence of events is very important in statistics and gambling. Whatever mechanism there is to ensure randomness has no memory whether it is a roulette wheel or two different competing terrorist groups trying to bomb the same plane on their own. Once the event is done, the same pool of winners and losers go back into the same statistical pool as everyone else. A few of those people go on to experience something unusual again and the universe of all ‘unusual things’ is huge and is only determined after the fact. That is where we get stories of people that witnessed multiple tragedies like ship wrecks or stumbled upon good fortune multiple times. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [May 24, 2011, 1:14am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/5 "2011-05-24T01:14:04Z")
> [@dracoi](#): > > To put numbers to the previous explanation: > > Chance of one bomb, normally: 1% > > Chance of two independent bombs at once: 0.01% > > Chance of two bombs, given that we already know one is there: 1% 1 in 100 planes have a bomb on them? ---
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [May 24, 2011, 1:26am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/6 "2011-05-24T01:26:21Z")
Just had to share this. A couple had two children. A friend asked them if they were planning a third. They said, “Well, we wanted three, but we heard that one out of every three children born in the world is Chinese, so…” ---
**Author:** ![Sage\_Rat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sage_rat/32/399_2.png) [@Sage\_Rat](https://boards.straightdope.com/u/Sage_Rat)\ **Post date:** [May 24, 2011, 1:31am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/7 "2011-05-24T01:31:16Z")
It’s called the [Gambler’s Fallacy](http://en.wikipedia.org/wiki/Gambler%27s_fallacy). Your odds aren’t any better or worse since what you do doesn’t affect what terrorists are going to do that day. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [May 24, 2011, 6:04am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/8 "2011-05-24T06:04:55Z")
> [@Sage\_Rat](#): > > It’s called the [Gambler’s Fallacy](http://en.wikipedia.org/wiki/Gambler%27s_fallacy). Your odds aren’t any better or worse since what you do doesn’t affect what terrorists are going to do that day. Thanks. In fact, the very joke in OP is cited in that Wiki article. **Shagnasty,** could you repeat/elaborate your post, including your “nitpick”? And, knowing what I know about the Gambler’s Fallacy, why would would the lottery buyer’s chances be “in favor” of winning a second time? Don’t all the odds reset and go back into the “chance has no memory” pot? Clearly I am missing a lot in reading your post. ---
**Author:** ![Malacandra](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Malacandra](https://boards.straightdope.com/u/Malacandra)\ **Post date:** [May 24, 2011, 6:08am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/9 "2011-05-24T06:08:31Z")
> [@TriPolar](#): > > 1 in 100 planes have a bomb on them? No, that figure was quoted to make the math easy. The real answer is left as an exercise for the student. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [May 24, 2011, 6:11am UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/10 "2011-05-24T06:11:10Z")
ETA after timeout: I just figured out “nitpick.” Read sentence structure wrong. As you were. 🙂 ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [May 24, 2011, 2:09pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/11 "2011-05-24T14:09:42Z")
Strictly speaking, the Gambler’s Fallacy is about multiple trials separated by time. The OPs case is really just a (mis-)application of Bayes Theorem (mentioned in passing in the wiki on Gambler’s Fallacy). Or more precisely, Bayes theorem describes the correct thinking which \*\*dracoi \*\*gave perfectly in post #3. The OP’s scenario demonstrates common naive non-Bayesian thinking and the errors which result from it. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [May 24, 2011, 3:04pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/12 "2011-05-24T15:04:05Z")
Also known as the [Baldrick’s bullet](http://mongers.org/jokes/black_adder_fourth) fallacy. 🙂 ---
**Author:** ![Ximenean](https://avatars.discourse-cdn.com/v4/letter/x/aca169/32.png) [@Ximenean](https://boards.straightdope.com/u/Ximenean)\ **Post date:** [May 24, 2011, 3:32pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/13 "2011-05-24T15:32:35Z")
I don’t see how **robert columbia** and **dracoi** ’s argument has much to do with Bayes therorem, which is about calculating “B given A” probabilities when you know “A given B”. Instead, the guy in the OP knows that P(A given B) is small, but assigns the wrong kind of event to B - a known bomb, rather than one he doesn’t know about. The known bomb is irrelevant to the conditional probability. ---
**Author:** ![Contrapuntal](https://avatars.discourse-cdn.com/v4/letter/c/e274bd/32.png) [@Contrapuntal](https://boards.straightdope.com/u/Contrapuntal)\ **Post date:** [May 24, 2011, 3:38pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/14 "2011-05-24T15:38:15Z")
> [@Colophon](#): > > Also known as the [Baldrick’s bullet](http://mongers.org/jokes/black_adder_fourth) fallacy. 🙂 Nice. ---
**Author:** ![Wheelz](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wheelz/32/5718_2.png) [@Wheelz](https://boards.straightdope.com/u/Wheelz)\ **Post date:** [May 24, 2011, 3:42pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/15 "2011-05-24T15:42:58Z")
This reminds of a scene from _The World According To Garp_. Garp and his wife are house-shopping. They are standing in the front yard of a house with a realtor, when a small plane comes buzzing over the horizon and crashes into the house, demolishing the entire second floor. Suddenly Garp exclaims, “We’ll take it!” In answer to the puzzled looks of his wife and the realtor, he explains, “The odds of another plane hitting this house are _astronomical_. We’ll be safe here!” ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [May 24, 2011, 3:51pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/16 "2011-05-24T15:51:09Z")
> [@Colophon](#): > > Also known as the [Baldrick’s bullet](http://mongers.org/jokes/black_adder_fourth) fallacy. 🙂 I’d bet _all_ logical fallacies could be re-attributed to Baldrick. And I would heartily approve of such a naming scheme. ---
**Author:** ![Ximenean](https://avatars.discourse-cdn.com/v4/letter/x/aca169/32.png) [@Ximenean](https://boards.straightdope.com/u/Ximenean)\ **Post date:** [May 24, 2011, 4:48pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/17 "2011-05-24T16:48:44Z")
> [@Ximenean](#): > > I don’t see how **robert columbia** and **dracoi** ’s argument has much to do with Bayes therorem, which is about calculating “B given A” probabilities when you know “A given B”. Instead, the guy in the OP knows that P(A given B) is small, but assigns the wrong kind of event to B - a known bomb, rather than one he doesn’t know about. I got muddled in the second part of that post - it’s not that he knows that P(second bomb given first bomb) is low; he knows that P(two unknown bombs) is low, and confuses that with P(second unknown bomb given one known bomb). As others have already said, of course. ---
**Author:** ![chappachula](https://avatars.discourse-cdn.com/v4/letter/c/d2c977/32.png) [@chappachula](https://boards.straightdope.com/u/chappachula)\ **Post date:** [May 24, 2011, 5:14pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/18 "2011-05-24T17:14:35Z")
> [@Shagnasty](#): > > What about someone winning the lottery twice, .. when you run the statistics of all lottery winners, **the odds are in favor of at least one of those prior winners winning again** just by chance > > Once the event is done, the same pool of winners and losers go back into the same statistical pool as everyone else. A few of those people go on to experience something unusual again and the universe of all ‘unusual things’ is huge. Huh? I need some explainin’ done, please. (in small words, please 🙂 ) I know why somebody wins the lottery every week; if the odds are a million to one, and a million people buy lottery tickets, then obviously _somebody_ will win. But the chances that \*\* I personally\*\* will win are vanishingly small\*, right? Now,let’s say that the next week, the odds are the same, and the public is the same–the million ticket buyers are the exact same people as the week before. Obviously, _somebody_ will win. But ,once again, the chances that \*\* I personally\*\* will win are vanishingly small. Now, let’s say I buy a weekly ticket all my life, and so I accumulate 2500 tickets (50 tickets a year for 50 years), each with a million-to-one odds against winning. And all the other ticket buyers do the same-- that’s a million people , each buying 2500 tickets —so 2,500 000 000 opportunities to win–but there is a total of only 2500 prizes, so the odds are still the same-- (i.e. vanishingly small) that \*\* I personally\*\* will win even once.And the likelihood of me winning twice seems even more vanishingly small. Yet the quote above says “one of those prior winners will win again, **just by chance** ”. What am I not understanding? \*that’s why I never waste the money on a lotto ticket. ---
**Author:** ![Jas09](https://avatars.discourse-cdn.com/v4/letter/j/d07c76/32.png) [@Jas09](https://boards.straightdope.com/u/Jas09)\ **Post date:** [May 24, 2011, 5:30pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/19 "2011-05-24T17:30:50Z")
> [@chappachula](#): > > And the likelihood of me winning twice seems even more vanishingly small. > > Yet the quote above says “one of those prior winners will win again, **just by chance** ”. > > What am I not understanding? There is a difference between _you_ winning twice and _anyone_ winning twice. ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [May 24, 2011, 5:33pm UTC](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952/20 "2011-05-24T17:33:56Z")
> [@chappachula](#): > > What am I not understanding? I think what you are missing is that we don’t care if any particular lottery winner wins a second time, just that one of the previous lottery winners wins a second time. You’re making the same mistake that the passenger in the OP is making by fixing one of the variables. There are lots of previous lottery winners; any one of them can get lucky a second time. It’s similar to the birthday problem. If you ask _one person_ in a class of 35 people what are the odds of someone else in the class having the same birthday the odds are pretty low. But if you ask if there are _any two people_ in the class that have the same birthday the odds are pretty high. [Next page](https://boards.straightdope.com/t/what-is-the-logical-statistical-switcheroo-behind-this-joke/582952.md?page=2)