Many of my blogs have focused on cognitive biases or systematic errors of judgment. In perusing the internet, I found another interesting example of why we should not trust our intuition.
A LiveScience article (“What is the birthday paradox?”) posted on July 30, 2022, authored by Charles Q. Choi asks, “how big does a group need to be for “birthday twins“ to exist?” That is, how many people must a group contain for there to be two people with the same birthday in that group?
First, many people will misread this question and think how many people must be in a group to have someone share their birthday. That is not the question: it is to share any date as the same birthday.
Assuming the question is read correctly, most people will still analyze the issue incorrectly because of faulty intuition. Typically, a person will start the intuitive analysis with the fact that there are 365 days in a year and so to have a 50% chance of having two people in the room with the same birthday, the group needs to be 182 people. Wrong! The calculation does not consider the notion of mathematical probabilities and how the odds will increase exponentially as the group grows in size.
In truth, the answer is 23 people. To arrive at this answer, disregard leap years and assume that all birthdays have an equal chance of happening. Then:
If you start with a group of two people, the chance the first person does not share a birthday with the second is 364/365. As such, the likelihood they share a birthday is 1 minus (364/365), or a probability of about 0.27%.
If you assume a group of three people, the first two people cover two dates. This means the chance the third person does not share a birthday with the other two is 363/365. As such, the likelihood they all share a birthday is 1 minus the product of (364/365) times (363/365), or a probability of about 0.82%.
The more people in a group, the greater the chances that at least a pair of people will share a birthday. With 23 people, there is a 50.73% chance, Frost noted. With 57 people, there is a probability of 99%. (Id.)
Known as the “birthday paradox,” this bit of counter intuitive thinking is another reason why we must be careful about our assumptions and reflective thinking. Such thinking can adversely affect one’s options for settlement especially if the payment of money is over time. The article provides another example:
“In exchange for some service, suppose you’re offered to be paid 1 cent on the first day, 2 cents on the second day, 4 cents on the third, 8 cents, 16 cents, and so on, for 30 days,” Frost said. “Is that a good deal? Most people think it’s a bad deal, but thanks to exponential growth, you’ll have a total of $10.7 million on the 30th day.” (Id.)
While most of us probably are not math mavens, at the same time, we need to check our assumptions underlying any potential deal. We should not let our reflective/intuitive thinking make the decision for us that the proposed deal is a “bad” deal or a “good” one. As evidenced above, the exponential growth of pennies a day can lead to millions of dollars in a month.
.. Just something to think about.
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