Understanding the Monty Hall Problem

The Monty Hall problem is a logic puzzle named for the host of the gameshow Let's Make a Deal. It's one of my favorite such problems, because it's an example of math completely contradicting my gut instinct. Even though I know the math says to do one thing, my gut consistently says the other -- and thus every time the problem is explained, I get a little window into my fallible human brain. In fact, I used to be so tied to my gut on this one that I would fail to explain the Monty Hall problem correctly when describing it to others, because I always assumed my gut instinct was actually correct. (Yes, I took Math for Liberal Arts Majors.)

Marilyn vos Savant posed the problem in one of her columns like so:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

I won't spoil the answer. Think on it for a moment. Do you switch or not? Now watch this AsapSCIENCE video for a brief explanation of what math says is the right thing to do...and what my gut says is crazy:

Having said all this, it's important to note that we have a shared assumption that a car is better than a goat. But if we're talking about one of those awesome unicorn goats, maybe you want that instead.

Also relevant: previous coverage of Monty Hall, plus an interview with the man. And although they're not strictly related, the Monty Hall Problem occupies the same space in my brain as The Abilene Paradox.

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Does Self-Control Deplete Over the Course of the Day? Maybe Not, Says New Study
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For months now, I’ve been trying to cut out sugar from my diet. I’ve read about all the ways my sweet tooth will be the death of me, and I’ve resolved to give it up. And yet, even as I write this, my long-term goal to eat healthy is losing out to my eternal desire to eat M&Ms at my desk. Is it because it’s the end of the day, and I’ve been trying to make choices for eight hours already? Or is it something else?

A new study in PLOS One pushes back on the popular theory known as "ego depletion," which hypothesizes that self-control is a finite resource that depletes throughout the day, much like energy levels. Instead, researchers from the University of Toronto and the learning technology company Cerego found that people's self-control depletes when it comes to doing one task for a long period of time, but that self-control fatigue isn't a factor when you're switching tasks. In other words, it's hard to say no to the box of cookies all day long, but saying no to the box of cookies won't impede other acts of self-control, like your ability to focus on your homework instead of turning on the TV.

The study used data from Cerego, which publishes online study materials, examining the study behaviors of two groups of college students using the Cerego system as part of semester-long psychology courses. The researchers looked at data from two groups of users, one group of 8700 students and one of almost 8800, focusing on how long they worked during each session and how well they performed at the memory tests within the curriculum.

If self-control really is a finite resource, it should be depleted by the end of the day, after people presumably have spent many hours resisting their first impulses in one way or another. But the researchers found that this wasn't true. Overall, students didn't do any better if they used the program earlier in the morning. Instead, performances peaked around 2 p.m., and people logged in to use the software more and more as the day went on, suggesting that the motivation to learn doesn't fall off at night (though that may also be because that's when college students do their homework in general).

However, mental resources did seem to be drained by doing the same task for a long period of time. The researchers found that after a certain point, students' performance dropped off, peaking at about 28 minutes of work. They made about 5 percent more mistakes 50 minutes into the session compared to that peak.

When it comes to the idea that we exhaust our store of self-control, the authors write, "the notion that this fatigue is completely fluid, and that it emerges after minutes of self-control, is under considerable doubt."

The notion of ego depletion comes from a 1998 study in which researchers asked participants to hang out in a room full of fresh-baked cookies, telling them to eat only from a bowl of radishes, leaving the cookies untouched. Then, those volunteers worked on an impossible puzzle. Volunteers who had spent time avoiding the delicious pull of cookies gave up on the mind-boggling task an average of 11 minutes earlier than a group of volunteers who were brought into the same room and allowed to eat as many cookies as they wanted. (Lucky them.)

Since then, the idea has taken off, leading to hundreds of subsequent studies and even influencing the habits of people like Barack Obama, who told Vanity Fair in 2011 that he only wore blue or gray suits in order to cut down on the non-vital decisions he had to make throughout the day.

This current study isn't the first to challenge the theory’s veracity, though. In 2016, a 2000-person replication study by some of the same authors (with scientists in 23 different labs) pushed back on the theory of ego depletion, finding that short spurts of self-control didn't have any effect on subsequent tasks. This study just adds to the evidence against the well-established idea.

So it's looking more and more like ego depletion isn't a good excuse for my afternoon vending-machine habit. Perhaps the true secret to excellent self-control is this: Just be a raven.

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Think Other People Have More Friends Than You? You’re Probably Wrong
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If you've ever felt bad about how small your social circle seems compared to everyone else's, fear not. A new study finds that most people overestimate how large the social groups of people around them are, according to Business Insider. In other words, people think others are way more popular than they actually are.

The study, published in the Personality and Social Psychology Bulletin by researchers at the University of British Columbia, explored the prevalence of this misconception among first-year college students. Because freshmen are just entering a new social environment, many are leaving their longstanding social circles behind for the larger, unfamiliar territory of college life. They might not have made very many friends yet, but it appears that most believe that their peers have.

The two experiments surveyed a total of almost 1500 students combined. In the first, almost 1100 second-semester freshmen were asked about the number of close friends and acquaintances they had made at school—distinguished by whether or not they confided personal problems in them or not—then to estimate how many friends the other first-year students had made in the same time period. Almost half the students thought that others had more close friends at school than they did, while just 31 percent estimated that they had more close friends at school than others did. The same went for the number of acquaintances they had. The students reported having an average of 3.6 close friends of their own, but thought that others had an average of 4.2 close friends.

In the second experiment, the researchers followed almost 390 students, divided into two groups, for two years, asking them the same questions as in the first experiment. They also asked what percentage of their total time they spent socializing with friends they made prior to coming to college as well as what percentage of time they spent socializing with other students they met at UBC. They estimated how much time others spent on the same activities, then completed questionnaires on their well-being, life-satisfaction, loneliness, and sense of belonging.

Again, most of the students thought that other people had more friends than they did, and estimated that their peers spent more time socializing with their new college friends than they themselves did. This misperception extended even to their specific close friends and acquaintances, who they believed spent more time socializing with their other new friends than they did. However, the more time the participant spent with said friends and acquaintances, the smaller the gap between perception and reality were. Importantly, people who believed that everyone else was more popular than they were reported lower levels of well-being and a lower sense of belonging.

This misreading of others' experiences may in part be due to the fact that a lot of social activities are very visible, whereas hanging out by yourself is, by nature, not. Eating with a bunch of people in the dining hall is a public activity that others can see, whereas few people see you studying alone in your room. "This could make it difficult for students to imagine the prevalence of their peers' solitary activities and therefore to over-rely on peers' publicly visible social activities to estimate their peers' social connectedness," the researchers write.

The study only examined the perceptions of young people who find themselves in a totally new social environment, but it's easy to imagine that the same misperception could exist outside of college, too. It's not the only misconception we tend to have about friendship, after all. In 2016, a study revealed a depressing stat: As many as half of your friendships might be one-sided, meaning you consider someone your friend, but they don't consider you theirs.

It turns out, when it comes to our social lives, most of us have no idea what's going on.

[h/t Business Insider]


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