A mathematician's 'seemingly simple but counterintuitive probability puzzle' sparks online debate



On January 30, 2024, Daniel Litt , an assistant professor in the Department of Mathematics at the University of Toronto, posted a probability puzzle on X (formerly Twitter). This probability puzzle has become a topic of discussion on the Internet.

Perplexing the Web, One Probability Puzzle at a Time | Quanta Magazine
https://www.quantamagazine.org/perplexing-the-web-one-probability-puzzle-at-a-time-20240829/

The problem Litt posted was: 'Suppose you have a urn containing 100 balls, n of which are red and 100-n are green, where n is uniformly distributed between 0 and 100. You randomly draw a ball from the urn and it is red. You then discard it. From the remaining 99 balls, what color ball should you draw next?'




In the survey conducted by Litt, 37.1% of users answered that the correct answer was to draw the green ball. Meanwhile, only 22.6% answered that the red ball was more likely, 20.9% answered that both were the same, and 19.5% answered that they didn't know.

It certainly seems like the probability of drawing a green ball next is higher than drawing consecutive reds. However, because the first ball was red, the number of remaining red balls is n-1, and the number of green balls is 100-n. By drawing the first red ball, the average value of n has increased, and there is a possibility that there will be a relatively large number of red balls among the remaining 99 balls. If you actually calculate using Bayes' theorem , the probability that the second ball will be red is approximately 67.57%. Therefore, the correct answer is that there is a 'high probability that the next ball drawn will be red.'

According to Litt, he liked mathematician George Roeser's solution the best. Specifically, he explained, 'Imagine the balls in the urn as 101 numbered balls in total, and remove one. Color the ball to the left of it green and the ball to the right red. Next, select a second ball at random. In this problem, you draw a red ball, so it will be placed to the right of the first ball. The next third ball to be selected will be one of the following: 'to the left of the first ball,' 'between the first and second balls,' or 'to the right of the second ball,' so the probability that the ball will be red is 2/3.'




Litt's post has generated a buzz online, with replies such as 'You're neglecting your research and thinking about puzzles.'




Regarding the reason why so many users gave the wrong answer, Litt said, 'People avoid risk in various ways and systematically overestimate or underestimate the likelihood of realistically unlikely events like this one. Therefore, I think they instinctively chose the wrong answer.' Regarding what prompted him to post probability puzzles like this one, he said, 'Although I am not an expert in probability theory, probability problems are very familiar to me, and sometimes the answers are cool, counterintuitive, or surprising. That's why I post example problems on X.'

'There's nothing more refreshing than presenting a multiple choice question where the majority of users give answers that are significantly worse than random chance,' Litt wrote.


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