I can't remember where I heard of this story on funding the research for the Manhattan Project. (It may be apocryphal.) In any case, this version is complete fiction.

The Twenty Third Person

History is full of great people at monumental events, changing the world. People like Napoleon at Waterloo or Paul Tibbets dropping the atomic bomb at Hiroshima. But, what about those other people, the ones who started a chain of events that led up to that moment.

It was the fall of 1939. The war in Europe had started. 
Germany
They would get the math wrong and decide that a nuclear reaction could not be made into a bomb.
and
Britain
The joint British and Canadian research project "Tube Alloy" was merged with the US "Manhattan Project" in 1943 under the Quebec Agreement.
were in a frantic race to develop the new “super weapon” — the atomic bomb. In the United States the matter did not seem as urgent.

The principle behind the atomic bomb is really very simple. It is an uncontrolled chain reaction started when a neutron strikes the nucleus of a U-235 atom releasing energy and more neutrons.

The hardest part is getting it all together.

“Hurry Mary Margret. You don’t want to be late for work.” (Only her Mum used her full name, to everyone else she was plain ‘Mary’.)

“Yes Ma, I’m coming.”

Mary had a typical middle class bedroom, in a typical middle class house, in a typical middle class neighbourhood. The afternoon light coming through the windows was filtered by the large oak tree in the yard and the slightly yellowing lace curtains. A pink and mauve diamond patterned wallpaper graced the wall. One bed, one dresser and a closet, nothing fancy, but, Mary liked it.

As she finished straightening the pink chenille bedspread her mind turned to the news from Europe. She hated the thought of war. Her parents were from England. Every evening they tuned in the BBC world service to get the latest news. It got worse every day.

“At least Charlie won’t be going to war.” she thought.

Charlie was Mary’s boyfriend. She was waiting for him to ask the question. She already knew what her answer would be.

Charlie worked for the Army, Mary still wasn’t sure what he did. After he graduated from MIT he was assigned to the US Army’s Manhattan Engineering district as a civilian adviser. (That was all he would tell her.) What she really couldn’t understand was, if he was in the Manhattan district, why did he work here in Boston? And, why was he so excited that she would be working at Dr. Sachs’s dinner party tonight?

Alexander Sachs
Dr Alexander Sachs
Brought to President Roosevelt's attention the possibility of an atomic bomb:
 was busy getting things ready for his dinner party. He liked to host dinner parties. It was a way for people to relax and get to know each other outside of work. Normally he didn’t allow any business talk at his parties. Tonight, tonight would be different.
Szilard
Leo Szilard
Conceived the idea of a nuclear chain reaction that would ultimately lead to the development of the atomic bomb. - Wikipedia
and
Teller
Edward Teller
"known both for his scientific ability and for his difficult interpersonal relations and volatile personality." - Wikipedia
would be there as would
Adamson
Lt. Col. Keith F. Adamson
Member of Advisory Committee on Uranium which controlled funding for nuclear research
. Somehow they had to get him to agree to funding for
Teller’s project.
To build a machine capable of producing enough U-235 to construct a bomb.

Once the material is assembled it is necessary to keep the neutron from starting the reaction too early. This is done by adding dampening material that acts like a buffer between the neutron and the nucleus.

There were twenty two people in the small conference room Sachs was using for his party. Mary was used to working in crowded rooms. Expertly she carried the trays of food from the kitchen to the buffet table. She didn’t know most of the people in the room. She knew Dr. Sachs, of course. Charlie had asked her to watch out for Lt. Col. Adamson. Mary spotted him in a corner talking with Edward Teller and someone she thought was Leo Szilard. (Charlie had said they were very important.)

“Look Teller, I don’t see how I can justify giving you $6,000 to see if something might happen.”

“With respect Colonel, you don’t seem to understand, I am not saying it might happen, I am saying it probably will happen. I just need to prove the numbers are correct.”

Adamson looked even more puzzled. He didn’t get to be on the President’s advisory panel for nothing. He always considered himself as above average intelligence. However, there was something about probability that he just couldn’t grasp.

“Alexander, Alexander come over here for a minute.” called Teller. “Perhaps you can explain probability to the Colonel.”

Alexander Sachs, was a born teacher. He enjoyed nothing more than explaining statistics and probability.

“Colonel, probability is just another way of looking at random events.” Sachs explained. “Take this coin. If I flip the once, there is a 50-50 chance of having it land heads up. But, if I called heads and it landed tails, I would be wrong. In that case, I would be wrong 100% of the time. But, if I flipped the coin a thousand times, and kept calling heads, I would be right close to half the time. Probability lets us make predictions based on a large number of tries.”

Adamson laughed. That's great if you are flipping coins, I have to make decisions  on thousands of  dollars. You have to come up with a better example than that.”

Sachs smiled. “I see your point. How about if we try to predict birthdays?”

“What do you mean by that?”

“Well Colonel, if you have twenty three people in a room, the chances are better than 50-50 that two of them would have the same birthday.”

Adamson roared. “You’re kidding me. What have you got here, twenty two people? . . . That’s close enough. If I asked them their birthday, you’re saying two of them would be the same?” 

Without waiting for an answer, the Colonel’s voice boomed across the room. He explained Sachs’ theory and said he was going to test it by having everyone in the room shout out their birthday.

Teller groaned. With twenty three people the odds were 50-50 that two people would have the same birthday. With twenty two people the odds were 46-54 against. He felt sick thinking that the future of his project, indeed the future of the world, might be decided on the equivalent of a flip of a coin.

With all the food on the buffet table, Mary stood at the back of the room to see the results of the Colonel’s test.

One by one, the men in the room called out their birthdays. There were no matches.

Adamson was gleeful. “Alex, [Sachs hated people calling him Alex] I hate to say this, but your theory is bunk. Ed, you are going to have to come up with something better to get your funding.”

Teller was visibly shaken and leaned against the wall for support.

“Once everything is ready, the dampening material is removed and the neutron is allowed to hit the nucleus.”

Caught up in the moment, Mary did something completely out of character. “Excuse me Colonel.” she called. “Excuse me, I am the twenty third person in the room, and I have the same birthday as that gentleman over there.”

Adamson wasn’t happy. He was still sceptical of Teller’s project, but grudgingly muttered. “OK, You'll get your money”

Her work finished Mary went back to the kitchen to rest. A year later she and Charlie would marry. They would move to a small town called Oak Ridge in Tennessee. Charlie would work at Clinton Engineering Works making material for the Manhattan Project and the atomic bomb.

On August 7, 1945 she would read her newspaper with horror, wonder and awe about the atomic bomb being dropped on a city called Hiroshima. 

She would never know her part in that world changing event.