Interesting Problem

Here is an interesting problem that doesn’t require crazy math skills:

I participate in a daily lottery by choosing a number from 1-100 and hope it hits. The odds of winning is 1%.

I bought a ticket, got lucky, and won today. I decide to keep playing one ticket every day until I win again, and then stop playing for good. What day am I most likely to end up stop playing (by winning the lottery that day, of course)?

  1. Tomorrow
  2. The day after tomorrow
  3. 50 days from today
  4. 100 days from today
  5. There is no difference, every day is equally as likely

This is not a trick question (assume unlimited funds, standard lottery, etc.), no need to consider the unusual; you just need to understand the question.

If you chose 3) or 4), you would be incorrect. If you chose 5), congratulations, you have good statistical sense, and are probably quite sure of your answer, but you are nonetheless wrong – just less wrong. The correct answer is 1), tomorrow.

Counterintuitive? Here’s why.

Yes, every day is equally likely to hit the lottery as another, namely, a 1% chance. Let’s say today is Sunday. Tomorrow (Monday) I have a 1% chance to win and stop playing. The day after tomorrow (Tuesday) also carries a 1% chance to win. However for me to stop playing on Tuesday, two things have to happen: I must win on Tuesday (1%), and on top of that, I must NOT have won on Monday (99%). So while the chance I win on Tuesday is 1%, the chance that I stop on Tuesday is not 1% but 0.99%, because I must not have already won on Monday. Each day after that, the chance that I stop on that particular day decreases accordingly, not because I’m less likely to win on that day, but because I cannot have already won any day before then. Therefore, the most likely day that I will end up stop playing is tomorrow, which carries a 1% chance. Every day after that carries a chance of less than 1%.

This is a good example of how our intuitions fail us. I stated clearly that it is not a trick question, but “you just need to understand the question”, and for good reason. The question asked was “what day am I most likely to stop playing?”, which most people immediately substituted for a much easier question, “what day am I most likely to win?”. There is a subtle but important difference, which is the hard-to-spot implied condition of previous losses. To stop playing on a day does not just mean you win that day, but more importantly it implies an exact sequence of lose-lose-…-lose-WIN!

The last option “There is no difference, every day is equally as likely” is so appealing because it is a true statement.  The statement just happens to be irrelevant to the question.  It’s a powerful technique, a mental sleight of hand, widely used by marketers, politicians, and spouses.

Pope Endorsing Violence?

First, I agree with Pope Francis on many things he says and does (unlike his predecessor), and the direction in which he is taking the church. However, he has expressed an opinion that I strongly disagree with, and think is inappropriate for someone in his position. According to this report:

Gesturing towards Alberto Gasparri, a Vatican official who was next to him on board the plane, he said: “If my good friend Dr Gasparri says a curse word against my mother, he can expect a punch on the nose”.

Throwing a pretend punch, the Pope said: “It’s normal. You cannot provoke. You cannot insult the faith of others.


To a layperson, it seems that he is saying that if Face-Punch-4e0e4c3b2acd1_hiressomeone insulted his mother, he would punch him in the nose. That sounds like the Pope is endorsing violence against one who is not violent, simply because his feelings were hurt. Perhaps recognizing this misstep, the Vatican spokesman tried to do damage control:

Obviously he wasn’t justifying violence. He spoke about a spontaneous reaction that you can have when you feel profoundly offended. In this sense, your right to be respected has been put in question.

Ah, the good old “figurative”, “that’s not what he meant despite what he said” defense.

And Rev. Robert Gahl tries to further weasel out by saying,

Francis didn’t say that HE would have punched his friend for insulting his mother. He said his friend could expect to be punched, given that he should know that he had crossed a moral line in lobbing the insult and should be more careful and courteous in not causing offense.

Let’s see if this makes any sense by taking the holiness out. Imagine a drug lord telling an associate, “if dat punk ever *expletive, present participle* comes into mah territory again, he can expect a bullet through his *expletive, present participle* head”, while simultaneously making a gun gesture with his hand, pressing it against someone’s forehead, and fake-pulling the trigger. The intention of the drug lord – violence – seems more than clear, and the method – perforation of bone and thinking organ with metal pellet – is not ambiguous. Whether he does it personally or not, one can reasonably conclude: the drug lord does not merely passively condone, but actively endorses the threat of violence, and is ready to act if needed.

The example clearly shows that the Vatican spokesman was correct. The Pope obviously wasn’t justifying violence. He was ENDORSING violence. Ideas should be judged on merit alone; the person who expresses them should be irrelevant. Even if you are the Pope.

The line of argument taken by the Vatican spokesman and Rev. Gahl reminds me of the former President Clinton and Ms. Lewinsky case. Mr. Clinton narrowly avoided successful impeachment and perjury, by exploiting the incompetence of the prosecutor and creative but tortuous logic. He questioned the meaning of “is”, and came to the conclusion that even though Ms. Lewinsky was fellating him, she had sexual relations with him, but not vice versa! According to the declaratory definition, maybe. That’s getting off on a technicality, a distinction without a difference. For non-lawyers, think about how absurd that is for a moment. If one is getting serviced orally and concludes that he is not in a sexual relation, then what, exactly, is the nature of the act? For hygiene? For the taste?

Rev. Gahl argued that the Pope never said that HE would personally commit violence, despite the Pope himself gesturing with a flying fist to the victims nose. Technically he is correct. One is left to ponder, if not by the Pope himself, then by whom exactly? The Holy Secret Service? A Papal hitman? The Holy Mafia? None of these scenarios are any better than the Pope taking a swing himself. FacePunchOne often overlooked but exciting possibility is, perhaps the Pope’s mother, unhappy at being offended, flies from Argentina to find the person whom she likely does not know, and punches him in the face for an insult she did not hear. This would make the Pope merely unable to stop violence from happening, however if that is what the Pope meant, he needs to work on his communication. In any case, we are talking about a technicality. When others are trying to get you out of trouble, and the best they can do is rely on a technicality, you are on thin ice indeed. Especially if you are the Pope.

Contrary to what one might think, and as Dr. Pinker has shown in his highly recommended book, violence has greatly declined. Violence should rarely, if ever, be the response to non-violence in a civilized society. We have been moving in the right direction. We teach our children to “use words”, instead of fists, when they disagree. In a fight, we punish the one who threw the first punch. Disappointingly, this could be the Pope.

Freedom of speech is a basic human right, under Article 19 of the Universal Declaration of Human Rights by the UN. Unless the speech present a “clear and present danger”, all forms of expression should be protected, as unsavory or hateful as they may be. The problem is not with expression itself. It is with those who resort to violence when confronted with ideas they find offending; and to a lesser extent, those who feel that this type of violence is justified.


Update: The Pope, perhaps realizing the problematic situation he has created, backtracked somewhat.

Cancer in the News

Two separate articles about cancer hit the news recently, both of which elicited a strong reaction. One was about the occurrence of certain types of cancer often being due to chance rather than environment or lifestyle, and the other about cancer being the “best” way to die.

Reactions to the first article range from “dangerous viewpoint” “irresponsible” “garbage in, garbage out” “nihilistic” “bad science” to “smoke away” “time to binge” “told you so”. The article was published by researchers at John Hopkins University in Science, a top-tier peer reviewed journal. Although that does not automatically mean that the study is correct, it does imply that the study has been reviewed and vetted by people a lot more knowledgeable than myself and the typical commenter. It is irresponsible to dismiss a study as “garbage in, garbage out” or “bad science” simply because the implications do not fit one’s worldview.

Speaking of implications, what exactly are they? The typical strawman conclusion is that a healthy lifestyle doesn’t matter. However, that is not at all what the study concluded. The study stated that about 1/3 of the cancers studied could be attributed to genes and environmental factors. It did not look at all cancers (notably breast and prostate cancers), nor at other medical conditions, such as diabetes, which are very much linked to lifestyle. Not everyone gets cancer (much less the subset studied), not all cancers are incurable, and not all deaths are due to cancer. The “dangerous viewpoint” seems to be a kneejerk reaction to an imagined conclusion, a simplistic caricature of what the study actually says. Even at face value, putting the many obvious benefits of a healthy lifestyle aside, isn’t it worth it to improve one’s chances, even if the potential is a fraction of 1/3?

Reactions to the second article seem even more extreme, and mostly negative. Dr. Smith uses Mr. Bunuel, a well-planned end-of-life cancer case, to argue that compared to the other ways of dying, cancer is preferable. It gives the patient an opportunity to reflect and wrap up. Dr. Smith did not say that cancer is good, or that dying is good, a concept many of the commenters seem to be unable to grasp; he did make a case that out of all the different ways to die, cancer is less bad than the others.

We all must face death, and not enough of us have thought about it as carefully as we should. It is an event that greatly affects many, something too important to ignore because of a visceral aversion.  It’s not like we’re getting out of this alive.

Let’s see if an analogy can help take the visceral reaction out of the equation. If one were to be downsized from a company, would it be preferable to be given a month’s notice so one could tie up loose ends and handover work in progress, to be fired on the spot and escorted to the door immediately, or to gradually have your salary cut and be abused by supervisors until you leave in disgust?

End-of-life is not so cut and dry. Cancers, treatments, and circumstances vary widely, and perspectives are very different. A doctor is trained to be detached out of professional necessity, and it seems to be this perspective that most of the commenters find offensive. The commenters seem unaware that Dr. Smith has likely seen more cancer deaths than them combined, and a view from that perspective is probably worth considering with one’s brain rather than dismissing with one’s gut. I largely agree with this first portion of his post – I would choose time and manageable pain over sudden death.

I cannot agree with the last sentence, however. He states, “…. and let’s stop wasting billions trying to cure cancer, potentially leaving us to die a much more horrible death”. Even in the context of his post, it is difficult to see how such a blanket statement could be considered reasonable. He specifically said not to waste money trying to “cure cancer” – not drugs with astronomical cost but merely extend suffering by a month or two, which would have been a more defendable position. Yes, a blog post is meant to be informal, but this is something that should have been taken out or properly qualified before hitting the “post” button.