# Alexandria Jones and the Birthday Surprise

[Photo by D Sharon Pruitt.]

[July 27th is Alex’s birthday, which she shares with Johann Bernoulli, an irascible mathematician from the late 17th century.]

The guests had gone. Alex and her family sat around the table, sharing the last tidbits of birthday cake and ice cream. Alex smiled at her parents.

“Thanks, Mom and Dad,” she said. “It was a great party.”

Maria Jones, Alex’s mother, leaned back in her chair. “I do have one more surprise for you, Alex. But you will have to share this one with the whole family.”

Leon groaned. “I know what it is: Let’s all pitch in to clean up.”

“That wouldn’t be a surprise,” Alex said.

## The Birthday Surprise

“Thank you for volunteering, Leon,” Maria said. “But the surprise is . . . we’re going to have a baby.”

Alex and Leon spoke all in a jumble. “What? A BABY?! You’re kidding! Really?”

Dr. Jones laughed at his children. “Yes, a baby,” he said. “You will have a little brother or sister on December 15th.”

Maria shook her head. “Not likely. It goes against the laws of nature for a baby to be born on its due date, so there’s no telling when ours will actually be born. I suppose there’s even a chance we’ll have a Christmas baby.”

## Figuring the Probability

“Cool!” Leon said. “Let’s see — there are 31 days in December, so the chance is one in 31, right?”

Alex nodded. “And 1/3 is 33%, so 1/31 would be about a tenth of that. There’s a 3% chance our baby will be born on Christmas.”

Maria stood up and began to stack the plates. “Well, there’s nothing wrong with your math,” she said, “but you’ve made an unwarranted assumption. Can you figure out what it is?”

Alex picked up a stack of cups and carried them to the sink, where Leon was running soapy water. “Oh, I think I know our mistake,” she said. “We assumed it was equally likely for the baby to be born on any day in December. That’s probably not right, huh?”

## The Law of Large Numbers

Leon frowned. “How else can we do it?”

“There is no way to calculate the probability of a baby’s being born on its due date or any other day,” Dr. Jones said. “You would have to gather as much data as you could find on when babies are born, and then use the Law of Large Numbers to estimate the chances.”

“That sounds like science, not math.”

“Sometimes,” Dr. Jones said, “there isn’t much difference.”

“You know, Jacob Bernoulli’s birthday was in December,” Alex said. “Wouldn’t it be funny if my baby sister was born on a Bernoulli birthday, too?”

“Baby brother,” Leon said. “There’s a higher probability the baby will be a boy.”

## The Birthday Equation

Dr. Jones pulled out some paper and a pencil. “If you want to play around with algebra, here is a formula: Let d = the days you are looking at and n = the number of people in your group. Then the probability that at least two of these people will share a birthday is:

$P = 1 - \frac {d!}{d^n \left( d-n \right)!}$

Alex jumped at the chance to escape from washing dishes. Taking the pencil, she said, “So if I had a group of people who were all born in December, d = 31 days. And my formula would be:

$P = 1 - \frac {31!}{31^n \left( 31-n \right)!}$

Dr. Jones nodded. “Now, how many people would you need in your group to have a 50% chance of shared birthdays?”

“That’s easy,” Leon said. “Half of 31 is 15.5, so you’d need more than 15 people.”

• Was Leon right?
(Hint: If he was right, would I bother to ask?)
• What is the probability that the baby will be born on the birthday of one of the mathematician/scientists listed below?
• Can you explain Dr. Jones’s birthday equation?
What assumption does he make?

## Birthdays in December

13 — George Pólya, who wrote Dr. Jones’s favorite book: How To Solve It
14 — Tycho Brahe, astronomer
22 — Srinivasa Ramanujan, number theorist
26 — Charles Babbage, first computer
27 — Johannes Kepler, astronomer
27 — Jacob Bernoulli, Johann’s older brother

## To Be Continued…

Read all the posts from the July/August 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.