The statistics of this problem suggest you are better off switching. The contestant owns a door with a 1/3 probability of containing the prize. The other two doors collectively hold the remaining 2/3 probability. Monte Hall's offer boils down to this: Would you like to hold the option on one, randomly chosen door or the other two? (Because he will reveal one of the remaining two to be a loser, the contestant gets the prize if is was behind either of the two non-chosen doors to begin with.)
Esther and Elin gave a nice demonstration about how the counter-intuitive nature of this result can be overcome by having students play the game repeatedly with playing cards. But what I am writing about today is a question they asked at the end: Can anyone give an example of the Monte Hall problem in real life. One of the session attendees threw out the academic job search. We typically bring in 3 candidates. We get a sense of who we like. Then one of the other two bows out due to exogenous reasons. Do we switch preferred candidates?
I'm in the middle of hiring and so have been thinking about this. My first thought has been that my process isn't really like the Monte Hall problem. I don't have a choice between one car and two goats. I have three goods of varying quality. And I am not randomly picking. I am assessing them and then sorting out an expected value. So, when one job candidate bows out, it is nothing like the choice: Two random doors or one?
However, suppose my objective function was simply that I had to get the best candidate. Perhaps I have a strong sense of regret and just can't stand to learn that I didn't pick the best colleague when I had the chance. My guess is that even though I do my best to assess candidates, many times I really can't rule out the possibility that any of the three would be the best of the bunch. So long as none of the candidates is so clearly dominant that s/he has a probability of more than 50% of being the best, I'm back in Monte's world. So, if we assume someone with this kind of objective, the Monte Hall problem does apply.
Curious what others think. Is your hiring practice similar to the Monte Hall problem? Either way, do you know of other "real" examples of the problem in everyday life?
"That's $500 on every single seat–not just in this stadium, but in every professional football stadium in America [with visual of an NFL stadium]."
We can certainly discuss whether this is the 'right' way to think about this problem:
- How big is $1B? $1B is 1/1000th of the approximately $1T paid in income taxes. Is 99.9% a bad success rate?
- We already spend 31.5 billion dollars to pay tax professionals and buy tax software. How much more will we have to pay to squeeze out that last $1B?
- Do we know that the $1B is really "lost?" I know that when I do my own taxes I am conservative at times, intentionally leaving some deductions off my taxes to reduce the odds of an audit. Of course, my hypothetical tax preparer doesn't have to worry about feeling the full brunt of that audit since she isn't on the hook if I say I did something that I actually didn't. So, her incentives are to push me to claim the biggest refund possible.
But, all of that stated, I think it is great to see advertisers who are QR-literate and use "compared to what" to make their pitches as clear as possible.
Those who regularly exercise their QR state of mind routinely find themselves asking, "Compared to what?" So, as I look out my window at daytime temps of -20o F (we're headed to a high of -15o F from a low below -20o F) I thought I would take on this important question.
- While Ohare Airport posted a record, meaning they haven't been this cold at this time of year in over 100 years, in Minnesota our -20o F won't even come near the coldest 10 observations. (You have to get to -33o F to make that list.)
- The South Pole clocked in at -11o F this morning. We could send them "Wish we were there!" postcards.
- Today's predicted windchills of -40o F to -50o F–or even the alarmist warnings of -60o F–won't top (or bottom?) the 1936 Twin Cities record of -67o F'.] Note: Prior to 2001 we reported that as -87o F, but then we figured out that the formula for calculating the effects of wind speed on perceived temperature were overly aggressive. We don't want to exaggerate! It was only -67o F!
- The [link http://usatoday30.usatoday.com/weather/resources/askjack/wasnow.htm 'average low temperature at the US South Pole Station during the months of July-September is -81o F. At that temperature, even a breeze of 10 mph creates a windchill below -100o F.
All in all, I feel better about my lot when it is put in perspective. Another advantage of QR literacy!
- This is a great example of taking QR seriously. In the context it's being used, the consideration of trade-offs seems pretty sound.
- Given that the research on the under-use of 4th down has been around a while and we still don't see many coaches following this strategy, there must be some other component to coaches decision-making. For example, could it be that to lose unconventionally poses a greater risk to job security than losing the same way everyone else loses? Or could it be that winning isn't the only thing (pace Mr. Lombardi)–that coaches are trying to give students a "football experience" even if that doesn't mean maximizing the probability of wins.
- Recent concerns surrounding head injuries have led some to question the danger of kickoffs and punt returns. This video suggests the game wouldn't necessarily be radically different at the high school level if rules were adopted to eliminate these risky plays. (For example, rule could heavily penalize any kickoff that traveled more than 20 yards and outlaw punts.) Presumably receiving teams would get better at handling "on side" kicks and, as the video points out, the change in starting field position wouldn't necessarily be that great.
Overall, I share Christopher's recommendation of the video as an interesting application of QR to sport.
Now, that's a pretty big number. But just how big is it? I poked around Nature's website and found this interesting blog that gave me an answer. Author Bradley Voytek notes that the most common comparison is that "there are as many neurons in the human brain as there are stars in the Milky Way." But Voytek tells us that this isn't quite right–our best guess for stars in the Milky Way is between 200 and 400 billion.
He also shares some details about how we estimate the number of neurons. (Turns out counting one by one isn't a good strategy–who knew?!) Problems abound. The brain isn't uniformly dense with neurons, so sampling matters. And the neurons are so intertwined that the current best estimate comes from a strategy of disolving brain samples and introducing a dye that sticks to the nuclei of neurons. All in all, a pretty interesting example for teaching ideas of representative sampling and estimation techniques. An interesting problem to which your students can apply a few of those neurons!