In this intriguing probability problem, your goal is to maximize your chance of winning a free movie ticket based on birthdays.

The Setup

  • A line forms at a movie theater.
  • The first person whose birthday matches someone earlier in line wins a free ticket.
  • You can choose your position in the line.
  • Everyone’s birthday is uniformly and independently distributed over 365 days.
  • You don’t know anyone else’s birthday.

Question:

Which position in line should you choose to maximize your chance of being the winner?

Key Insight

You want to be the first person whose birthday matches someone earlier in line. This is related to the classic birthday paradox, but with a twist: you win only if your birthday matches someone before you, and you are the first such person.

Probabilistic Structure

Let’s define:

  • For position \( i \), the probability that no one before has your birthday is:

\[ P(\text{no match before}) = \left( \frac{364}{365} \right)^{i-1} \]

  • The probability that someone before you has your birthday is:

\[ 1 - \left( \frac{364}{365} \right)^{i-1} \]

  • But for you to win, no one before you can have matched someone before them—you must be the first repeater.

So we compute:

\[ P(i \text{ wins}) = \left[\text{probability no repeats among first } i-1\right] \times \left[\text{probability person } i \text{ matches someone before}\right] \]

This is maximized at position:

\[ \boxed{20} \]

Conclusion

If you can choose your place in line, position 20 gives you the highest chance of being the first person with a matching birthday and winning the free ticket.

Reference