# The M1M1 treasure hunt, week 4

# Chapter 2: Talkin' 'bout my generation

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Ann knocked once more on the door of the strange blue box.

"Come in. Well? What's the maximum value of a?" demanded her tutor.

"Er, 3, I think," she essayed uncertainly.

"Wrong. Thank goodness. I thought you might have decided to work really hard to get it right even though I told you last week you'd got it wrong, and so upset the fragile nature of Causality. We time travellers have to be very careful."

"I did try!" Ann protested. "But I'm worried that if I travel into the future and find all the answers, won't that count as plagiarism?"

"Hmm. I suppose that could be a problem. Maybe we should stick with the past for now. When would you like to go?"

"I'm ready now," she answered eagerly.

"NO, NO, NO. When IN THE PAST would you like to visit?" snapped the irrascible Dr Hu.

"Oh, I see. Well, I'm quite keen on M1S1. Could we travel to the birth of Statistics?"

The doctor made a few calculations, pressed some buttons, pulled some levers and turned on the light switch. Tentatively Ann opened the door.

"It's a girl, Mrs McCoy!" exclaimed a voice outside. And indeed, in front of her, Ann could see a baby girl, precociously sucking her thumb. As Ann watched, the baby removed her thumb and stared at her hands, as if trying to calculate in how many different ways she could insert N fingers from one hand and M from the other into her tiny mouth, with or without replacement.

"I didn't mean a real birth - I meant the beginning of Stats as a subject!" hissed Ann, more than a little embarrassed.

"Oh," mused the doctor, quietly shutting the door to the hospital room. "You mean like where does Statistics come from? What generates probability? Well, you need a probability generating function, of course. Functions define everything. Choose a positive integer at random."

"I can't do that!" cried Ann. "There's an infinity of them."

"Yes you can - they don't all have to have an equal chance.
Suppose you choose the integer, n, with probability p_{n}. Then we can define the function:

f(x) = ∑p_{n}x^{n}

That tells you everything you need to know. Suppose f(x) = 4/(10 - 7x + x^{2}). Then what's the probability p_{2} you'll choose 2, (written e.g. as 0.234)?"

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