The Birthday Paradox
How large does a random group of school children have to be for there to be a 50% chance that at least 2 of the children will share a birthday?
Answer: 23.
Did you miss the answer by a mile? Most people do.
Take a classroom of school children, for example. Let's say there are 30 children in the class who have 365 possible birth dates in a calendar year. The odds that any of the students would share a birthday seem pretty low, right? After all, in a group of only 30 children, whose arrivals were randomly spread over 10 times as many days throughout a year, none would probably share a birth date, right?
So, just how large does a group of random children need to be in order for 2 of them to share a birthday? Most people who quickly do the mental math will believe 182 is the correct answer, which is roughly half the number of days in a year. But would you really need 182 children in a group for 2 of them to have the same date of birth?
Answer: No. You will need a smaller number of people than 182.
People significantly underestimate how quickly the probability increases with group size. The number of possible pairings increases exponentially with group size. However, humans are not that great at guessing probabilities, especially when they are as counterintuitive as the birthday paradox
The Birthday Paradox
For a group of 2 school children, for example, the chance that one child will share a birthday with the other is 364 out of 365 days. This is a probability of about 0.27%. Add a third child to the group, and the chance of sharing a birthday shifts to 363 out of 365 days, which is a probability of about 0.82%. For simplicity, leap years, twins, selection bias, and seasonal and weekly variations in birth rates are generally disregarded, and instead it is assumed that there are 365 possible birthdays, and that each child's birthday is equally likely to be any of these days, independent of the other children in the group.
As you may have already guessed, the larger the group, the greater the odds that 2 children were born on the same day. So what is the right answer to the birthday paradox? If we keep doing the math, we'll discover that when we reach a group of 23 children, there will be about a 50% chance that 2 of them will share a birthday. At a group of 30 children there will be about a 70% chance that 2 of them will share a birthday, similarly at 40 children there will be about a 89% chance, at 50 children a 97% chance, at 60 children a 99% chance and then it drags with probability increasing by decimals till we reach a size of 366 children at which point there is 100% chance!
Why do 23 children seem like such a counterintuitive answer?
It all has to do with exponents. Our brains don't generally calculate the compounding power of exponents when we do the math in our heads. We tend to think that calculating probabilities is a linear exercise, which couldn't be further from the truth.
In a room with 22 other children, if you compare your birthday with the birthdays of the other 22 children, it would make for only 22 comparisons.
But if you compare all 23 birthdays against each other, it makes for many more than 22 comparisons. How many more? Well, the 1st child has 22 comparisons to make, but the 2nd child was already compared to the 1st child, so there are only 21 for that child to make. The 3rd child then has 20 comparisons, the 4th child has 19, and so on. If you add up all possible comparisons, the total is 253 comparisons, or comparison combinations. Thus, an assembly of 23 children involves 253 comparison combinations or 253 chances for 2 birthdays to match which is far more than half the number of days in a year.
What are the odds you plan to vacation in Paris this summer?
While probabilities and combinations may not be everyone’s cup of tea, a vacation is, right?
Talking about chances, what is the probability that you are worried about your upcoming international vacation? Doesn’t being worried about an upcoming vacation seem like a paradox? If Paris is the destination, then the probability is high that you have reason to be worried. Protests have erupted in dozens of French cities following the fatal shooting in late June 2023 of a teenager in a Paris suburb, resulting in riots and widespread destruction. Over 800 people were recently arrested, and more than 40,000 police officers have been deployed across the country to bring the situation under control.
With the peak summer travel season underway, many tourists are headed to Paris and other parts of France for their summer vacations. Exercise increased caution in France due to terrorism and civil unrest – is the general warning given by various consulates. In this light thought of cancelling or postponing your trip to Paris may have crossed your mind. Last-minute disruption of your travel plans might be difficult to navigate. From rebooking and refunds to finding the right airline contact details, these are your next steps.
Travel Insurance - Trip cancellation and Curtailment cover
What happens to the money that you have already spent to make arrangements for your trip? With a reasonable travel insurance policy, you will get reimbursed for all prepaid and non-refundable travel expenses.
What happens if you already embarked on the trip and are in the middle of it, you have to cut it short due to an unavoidable emergency? With trip curtailment cover that many insurers offer, you get protection against such a scenario as well. Not only the insurance policy would reimburse you for the hotel reservations and other prepaid expenses, it would also cover the cost of your return flight that you may have to book on short notice and on exorbitant prices. Also, if, due to an unforeseen event, your flight departure is delayed beyond a stipulated reasonable amount of time, the insurer compensates you for the same.
Another instance where a travel insurance policy helps is when you are taking connecting flights to your destination, and you miss one of these flights due to unavoidable reasons. Travel insurance covers the cost of the new flight you would have to book in place of the one that you missed if it’s part of your coverage.
The purpose of insurance is to cover an uncertain event. By taking an insurance cover you are covering the probability of uncertain disruption of your vacation plans and the financial implications that follow. Vacations are times to enjoy, an insurance cover any chances that it does not stay that way or at least compensates in financial terms if it doesn't.
Source: howstuffworks.com, wikipedia.org, france24.com, forbes.com
Author: Ashish Joseph George, MMS, CFP. The views shared in this article are my personal views and don’t reflect the views of any organization. This is not an investment advice
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Awesome column Ashish! Keep up the good work!