What is the minimum number of students for a likely win in a birthday bet?

In summary, the conversation discusses the minimum number of students in a class for a lecturer to win a bet about shared birthdays. The lecturer proposes using a Poisson formula to calculate the probability, and it is mentioned that the answer is 23. The conversation also includes a simplified version of the problem and a suggestion to continue the calculation until the chance of no shared birthdays falls under 50%.
  • #1
babtridge
16
0
Hi there, I'm a bit stuck and was hoping somebody could give me a couple of pointers...

A lecturer wages that at least one pair of students in his class have birthdays on the same day. What is the minimum number of students in his class for him to be likely to win the bet?

I have assumed a Poisson formula for Pm = (n^m).(e^-n)/m!
for the probability of m pairs having their birthdays on the same day, when n is the mean number of such pairs.

I know the answer is 23 but I am really struggling to obtain this.
Any pointers would be much appreciated. Cheers
 
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  • #2
(To greatly simplify the problem, I'll assume February 29th is a myth.)

With 1 person in the room, there is a 100% chance that there would be no shared birthdays.

With 2 people, the chance of having no shared birthdays is 364/365. With 3 people, the chance is 364/365 * 363/365.

Continue this and you should get the answer. Remember, you're looking for the chance of having no shared birthdays to fall under 50%.
 
  • #3
OK, that's nice and clear now. Thanks for your reply CRGreathouse!
 

FAQ: What is the minimum number of students for a likely win in a birthday bet?

What is the Poisson formula and how is it used?

The Poisson formula is a mathematical equation used to calculate the probability of a specific number of events occurring within a given time period, assuming that the events happen independently and at a constant rate. It is often used in fields such as statistics, biology, and physics to analyze data and make predictions.

What are the components of the Poisson formula?

The Poisson formula consists of three main components: the mean rate of occurrence (λ), the number of events (k) that are being calculated, and the base of the natural logarithm (e). These components are combined in the formula λ^k * e^(-λ) / k! to calculate the probability of k events occurring.

How is the Poisson formula different from other probability formulas?

The Poisson formula is unique in that it specifically calculates the probability of a certain number of events occurring, rather than the probability of a specific outcome. It also assumes that the events are independent and occur at a constant rate, unlike other formulas that may take into account factors such as dependent events or changing probabilities.

Can the Poisson formula be used for continuous data?

No, the Poisson formula is only applicable to discrete data, meaning data that can only take on specific, separate values. Continuous data, such as measurements or time, cannot be used in the Poisson formula as it requires a whole number value for the number of events (k).

What are some real-world applications of the Poisson formula?

The Poisson formula has many applications in various fields, including predicting the number of accidents or natural disasters in a given time period, estimating the number of customers at a store, analyzing the spread of diseases, and predicting the number of mutations in DNA. It is also commonly used in quality control and inventory management to determine the probability of defects or stock shortages.

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