Mind-Bending Probability Questions

Watch as your mind is thoroughly broken by these easy to state probability questions.

intuiting probabilities = too hard for us mortals

Problem 1 – That Tuesday Boy Is Your Bane

Ajax has exactly two children. For no reason, in particular, you decide to ask Ajax, “is at least one of your children a male that was born on a Tuesday?” and he truthfully replies, “yes”. Given this information you just learned, what’s the percentage chance that both of Ajax’s children are male (assuming a random child born in Ajax’s city has a 50% chance of being male)? [1]

Hint: it’s not 25%, or 50%, or 66.6%, or 75%.

Problem 2 – Even Numbers Will Conditionally Drive You Crazy

What is the average number of times that you must roll a six-sided die (with sides labeled 1 through 6) before you roll a 6 (including the time you rolled the 6), conditioned on the fact that (i.e., assuming we are in a world where) all the die rolls ended up giving even numbers? [2]

Hint: it’s not three rolls, and it’s definitely not six rolls

Problem 3 – Your Study Can’t Even Tell Me Nothing

Suppose that Miggy calculates the correlation between two variables using data she’s collected and computes that the value is 0.0. Assuming that the data has no outliers, approximately how many data points would Miggy have needed to have used in the calculation such that the 95th percentile confidence interval for the correlation is -0.1 to 0.1 (that is, such that she can be reasonably confident that the correlation is not smaller than -0.1 or bigger than 0.1 in a 95th percentile confidence interval sense, and hence can conclude the true correlation is likely to be close to the 0.0 correlation that she calculated)? [3]

Hint: it’s more than 63 data points

Problem 4 – Your 99% Accuracy is 99% Worthless

Suppose that a test for a rare disease is 99% accurate (i.e., it says you do have the disease 99% of the time if you actually do have it, and says you don’t have the disease 99% of the time if you actually don’t have it). If only 1 in 10,000 people in the population have the disease, and the test gives a positive result for Joe (who was screened as part of a completely random screening), what’s the percentage chance that Joe actually has the disease? [4]

Hint: it’s less than 99%

Problem 5 – It’s Not a Paradox, It’s Just What It Feels Like For Our Brains To Suck At Probability

50 random people are in the same room. What’s the percentage chance that two or more people in that room share the same birthday (i.e., the chance that they were born on the same day of the year but not necessarily born in the same year). [5]

Hint: it’s a lot more than 50/365 turned into a percentage (which is about 14%)

Problem 6 – You Can’t Even Tell How To Tell Why Your Lawn Is Wet

Cadence’s lawn sprinkler is set to water the lawn a random 50% of mornings (completely independently of the weather). Cadence sleeps late and wakes up to find that the lawn is wet (which could have been caused by it raining, or by the lawn sprinkler going off that morning, or by morning dew from condensation, or perhaps multiple of these). Her goal is to figure out if it rained that morning or not. She’s about to go inspect the sprinkler to try to confirm if it went off that morning, but then her annoying boyfriend Kelby interjects and reminds her that whether the sprinkler goes off on a given morning is completely independent of whether it rains, so checking the sprinkler provides no indication as to whether rain occurred. Is Kelby right? [6]

Hint: can statistically independent events become correlated when you learn information or is that impossible?

Problem 7 – Some Of Your Bias Is Social; The Rest is Mathematical

Admission figures in 1973 for Berkeley graduate school showed that 44% of male applicants were admitted but only 35% of female applicants (a difference large enough that it was very unlikely to be due to chance or sampling error). Yet when looking at individual departments one by one, it became clear that male and female applicants were admitted at basically the same rates into the individual departments, and if anything, the individual department figures suggested that departments were slightly more likely to admit women than men. How is this possible? And was Berkeley actually showing a gender bias against women in their graduate admission during that period? [7]

Hint: there are other subtle factors that can vary by gender

Problem 8 – Proof That Math Makes You Unpopular

For the significant majority of people, the average number of Facebook friends that their Facebook friends have is higher than the number of Facebook friends that they themselves have. Worse still, it’s not just true of Facebook friends: it’s true of real-life friends as well. How is this possible? [8]

Hint: what sort of person is a person more likely to be friends with?

Problem 9 – Your Brain’s Even Less Trustworthy Than a Game Show Host

You’re on a game show, and one of the three closed doors in front of you has a golden goat behind it. The other two each have goats made of lead. Regardless of which door you pick, the game show host will then open one of the doors that you didn’t pick to reveal a lead goat. You then have the choice of staying with your original door you picked or switching to the other door that the game show host has not yet revealed. If you’re trying to get the gold goat, does it change your probability to switch doors? If so, should you switch or stay with your original guess if you want to maximize your chances? [9]

Hint: switching does change your chance of getting the gold

Problem 10 – Reclassifying People Can Be Good For Their Health

For internal bookkeeping purposes, a hospital divides patients into two groups based purely on each patient’s diagnoses: “serious illness” and “mild illness.” The hospital starts an initiative to improve the accuracy of their diagnoses, and discovers that some people were put in the wrong category, and moves them to the correct category. They are shocked to discover that after moving these patients to the proper classification, the average survival rate for both categories of patients immediately goes up! “Woohoo!” they shout in unison, “the investors will love this!” How is it possible that the average survival rate of both groups went up simply by moving patients from one group to the other? [10]

Hint: remember that the average survival rate of one group started much lower than the other since the patients were classified by the seriousness of their illnesses

[1] https://www.jesperjuul.net/…/tuesday-changes-everything-a-…/ [Answer=48%, that is, 13 out of 27. It has to do with “being born on a Tuesday,” likely only applying to one (not both) of the children, which means that the Tuesday bit actually gives you a slight amount of information that (probably but not necessarily) uniquely identifies one of the sons.]

[2] https://gilkalai.wordpress.com/…/elchanan-mossels-amazing-…/. [Answer=1.5 rolls on average. It has to do with the evenness of the six not mattering; it doesn’t matter what number you end up as long as it’s not one of the other numbers you’ve conditioned on. It’s very tricky.]

[3] http://vassarstats.net/rho.html [Answer=380. People’s intuition tends to be that we need a lot less data than we really do (part of the reason why scientists often use sample sizes that are too small). Correlations illustrate this problem to an especially large extent.]

[4] https://betterexplained.com/…/an-intuitive-and-short-expla…/ [Answer=about 1%. Most of the people who test positive out of the 10,000 will be false positives (about 100 false positives, whereas there will only be one true positive; hence there is only a 1 in 101 chance that the person actually has the disease.)]

[5] https://en.wikipedia.org/wiki/Birthday_problem [Answer=97%. Imagine the people entering the room one by one and checking whether their birthday matches anyone who is in the room so far. When the second person enters, there is only a 1 in 365 chance of having the same birthday as the person already in the room, but when the third person enters, there is a 2 in 365 chance, then 3 in 365 with the fourth person, and so on. With each new person, the chance of a collision increases, making the chance that any collisions occur very high once you’ve done this 50 times. It’s like shooting arrows randomly at a target, but the target grows after each miss.]

[6] https://en.wikipedia.org/wiki/Berkson%27s_paradox [Answer=No, Kelby is an asshole. While the sprinkler and the weather are statistically independent when you don’t know any other information, once you condition on the fact that you know the lawn is wet (that is, that it either rained or the sprinkler went off), that destroys the independence of the events and makes them negatively correlated.]

[7] https://en.wikipedia.org/wiki/Simpson%27s_paradox… [Answer=No, it seems that in this specific case, women were just more likely to apply to more competitive departments than men, and hence had lower admission rates (as anyone would if they applied to those more competitive departments).]

[8] https://en.wikipedia.org/wiki/Friendship_paradox [Answer=the probability you are friends with someone is higher if they have lots of friends than if they have only a few friends, so people have a biased selection of friends (which is more heavily weighted towards those with lots of friends)]

[9] http://gizmodo.com/heres-the-best-explanation-of-the-monty-… [Answer=you should switch. You can think of switching as having the opportunity to bet on both of the doors that you didn’t pick on your first guess since the host simply pointing out that one of the two other doors has a lead goat doesn’t change the probability that either of those two doors has the gold goat.]

[10] https://en.wikipedia.org/wiki/Will_Rogers_phenomenon [Answer=some of the sickest people who had previously been classified under “mild illness” probably got reclassified to having a “serious illness,” hence the “mild illness” group lost some of its sickest people (increasing its average survival rate), but those people who were moved into “serious illness” were less sick than the average for that “serious illness” group, so the “serious illness” group also had its survival rate increase.]