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The Birthday Problem

The Birthday Problem. The Problem. What is the probability that at least two people in this class share the same birthday?. Assumptions. Only 365 days each year. Birthdays are evenly distributed throughout the year, so that each day of the year has an equal chance of being someone’s birthday.

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The Birthday Problem

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  1. The Birthday Problem

  2. The Problem • What is the probability that at least two people in this class share the same birthday?

  3. Assumptions • Only 365 days each year. • Birthdays are evenly distributed throughout the year, so that each day of the year has an equal chance of being someone’s birthday.

  4. Take group of 5 people…. Let A = event no one in group shares same birthday. Then AC = event at least 2 people share same birthday. P(A) = 365/365 × 364/365 × 363/365 × 362/365 × 361/365 = 0.973 P(AC) = 1 - 0.973 = 0.027 That is, about a 3% chance that in a group of 5 people at least two people share the same birthday.

  5. Take group of 23 people…. Let A = event no one in group shares same birthday. Then AC = event at least 2 people share same birthday. P(A) = 365/365 × 364/365 × … × 343/365 = 0.493 P(AC) = 1 - 0.493 = 0.507 That is, about a 50% chance that in a group of 23 people at least two people share the same birthday.

  6. Take group of 50 people…. Let A = event no one in group shares same birthday. Then AC = event at least 2 people share same birthday. P(A) = 365/365 × 364/365 × … × 316/365 = 0.03 P(AC) = 1 - 0.03 = 0.97 That is, “virtually certain” that in a group of 50 people at least two people share the same birthday.

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