Jimmy and the Boy or Girl Paradox

It was a hot, muggy day. Students were languidly completing a worksheet on probability.

Jimmy was fidgety, looking for an escape clause.

“Sir,” Jimmy announced, “I know that probability depends on grammar.”

The class looked up. They wanted a distraction. And also to find out what Jimmy was talking about.

I straightened up in my chair.

“I wasn’t aware grammar is the central part of a mathematics problem,” I stated.

“It isn’t,” Jimmy said. “But it should be. I’ll explain it.”

Jimmy came to the front and turned towards the class.

“A family has two children and at least one of them is a boy.

What is the probability that both children are boys?”

There was silence for a time.

“One half,” Sarah replied immediately.

“I agree,” said Michael.

“So do I,” Ben added.

Several others nodded.

“I don’t think it’s one half,” Emily said.

“Why not?” Sarah asked.

“I don’t know,” Emily admitted. “It just sounds like one of those questions where something is hiding.”

“So what’s the answer?” Michael asked.

“I think it’s one third,” Jimmy announced.

The room’s dynamics suddenly shifted.

Students were directly challenging each other about the outcome.

“No way!”

“Impossible!”

“That makes no sense!”

“It has to be one half.”

“I think Jimmy’s trying to trick us.”

Within seconds the class had split into two camps.

One side insisted the answer had to be one half.

The other, mostly Jimmy’s friends, wasn’t convinced, although none of them could explain why.

“How did you work out the answer?” I asked.

Jimmy hesitated.

“The internet.”

“Do you know why it’s one third?”

“No.”

“Then why do you believe it?”

“Because if the internet says it’s one third, and I say it’s one half, one of us is about to be embarrassed.”

Jimmy looked directly at me.

“And I know it won’t be you,” Jimmy concluded.

A round of laughter.

At this stage, someone had to be the judge.

I wrote on the board:

Boy, Boy

Boy, Girl

Girl, Boy

Girl, Girl

“Assume boys and girls are equally likely,” I explained.

“We are told that at least one child is a boy.”

I crossed out:

Girl, Girl

“Because that combination is impossible,” Jimmy explained.

“This is what remains,” I continued.

Boy, Boy

Boy, Girl

Girl, Boy

“How many possible outcomes remain?”

“Three,” Emily answered.

“And how many contain two boys?”

“One.”

The room became quiet.

“It really is one third,” Sarah stated.

Michael frowned.

“I still don’t like it.”

Jimmy nodded.

“I know exactly how you feel.”

“The children haven’t changed,” Michael said.

“No,” I replied.

“The parents haven’t changed.”

“No.”

“The family hasn’t changed.”

“No.”

“So how did the probability change?”

“Because probability depends not only on what is true,” I said, “but also on what information you are given.”

Jimmy looked unconvinced.

“I still think mathematics is playing word games.”

“Let’s change just one sentence,” I said.

I rubbed out the board.

“This time the parent says:

‘My eldest child is a boy.’

Now what?”

I wrote:

Boy, Boy

Boy, Girl

“The eldest child is already known to be a boy,” I explained.

“So there are only two possibilities.”

“And one of them has two boys,” Sarah said.

“So the probability is…”

“One half,” everyone replied simultaneously.

Jimmy stared at the board.

“The children didn’t change.”

“They didn’t.”

“The parents didn’t change.”

“They didn’t.”

“The only thing that changed was the sentence,” Jimmy admitted.

“Exactly.”

“Sir, that was hard.”

“It could have been harder.”

Jimmy looked worried.

“How?”

“Suppose there were three children, and you were told that at least two of them were boys.”

Jimmy froze.

“That wasn’t very nice. I’ve decided something.”

“What?”

“From now on, whenever anyone asks me a probability question…”

He paused for dramatic effect.

“I’m going to ask them to repeat it very, very slowly.”