*n*random people. In my second post, we expanded that to look at a more general problem of having a birthday within

*k*days time of another birthday in a group of

*n*random people.

This time, we'll be looking at the problem of determining which person in a list is most likely to share a birthday with the people before him. In other words, if randomly selected people enter a room one at a time, which person is most likely to be the first one to share a birthday with someone already in the room? Perhaps you'd like to know because you're looking for a gambling opportunity, betting your friends on which person arriving at a party will share a birthday with someone already there.

The probability of there being a shared birthday among a randomly selected group of

*n*people can be calculated from the expression below:

Equation 1 |

Suppose now we add one more person. The probability of there being a shared birthday in a group of

*n+1*people is:

Equation 2 |

The incremental change in the probability of there being a shared birthday in a group of

*n*people versus*n+1*people is the probability that the (*n+1*)th person is the first person to share a birthday with someone already in the group. So subtracting the first expression from the second gives us the probability that the (*n+1*)th person added to a group of*n*people will share a birthday with someone already.Equation 3 |

Now if we want to know what value of

*n*maximizes this probability, we can plot the function to find out. I've done that here:
The peak is at

*n*= 19. So, at 3.23%, the 20th person to enter a random group of people has the greatest chance of being the first one to share a birthday with someone already in the group.
Suppose now we repeat this analysis for the more general case of birthdays within

*k*days time of each other. The probability of there being birthdays with*k*days of each other among a group of*n*people is:Equation 4 |

And in a group of

*n+1*people, the probability is:Equation 5 |

Just as before, we can subtract

*Equation 4*from*Equation 5*to get the probability that the (*n+1*)th person is the first person with a birthday close to one already in the group:Equation 6 |

Plotting this beastly looking function will allow us to see which person entering the room is most likely to be the one having a birthday close to a birthday of a person already at the party.

The table below summarizes the information from all the peaks in the above figure.

There you have it. As random people enter a room, the 3rd is most likely to be the first to have a birthday within a month of a person already at the party. The 6th is most likely to be the first to have a birthday within a week. The 12th person is most likely to have a birthday within one day, and the 20th is most likely to have the same birthday as someone already at the party.

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