Why you don’t add across probability trees – Secrets of the Mathematical Ninja

"... Evidently not," said the student, with a look of sheer terror that was music to the Mathematical Ninja's eyes.

He smiled a nasty smile. "No," he said, "you categorically do not add probabilities as you go through the tree."

"You... multiply?" The student was cautious. The Mathematical Ninja hadn't actually inflicted any pain on him, but he wasn't sure whether that was by design1. "So the probability of the actor picking a blue scarf and a green hat is the probabilities multiplied together."

"Yes - they're independent, so you can do that. But why do you multiply rather than add?" There it was, the nasty glint again.

"Because... them's the rules?" The student gritted his teeth and looked nervous. "I don't know why."

The Mathematical Ninja nodded. "It's simple," he said. "If the actor only picks the blue scarf 50 times out of 100, he can't pick the blue scarf and the green hat 75 times out of 100. That doesn't make sense - he has to wear the combination less often than its constituent parts."

"And if you add the probabilities, you end up with higher probabilities rather than lower ones. But... doesn't multiplication make things bigger, too?"

"In some contexts," said the Mathematical Ninja. "When you multiply a positive number by 2, sure - things get bigger. But when you multiply by 1, they stay the same, and when you multiply by 0, you get 0. It stands to reason that if you multiply a positive number by another positive number less than one..."

"And all probabilities are positive numbers less than 1!"

"Non-negative numbers no bigger than 1," said the Mathematical Ninja, sternly - "when you multiply like that, you get a smaller probability, which is what you want."

"Wouldn't it be clearer to start your tree with some mahoosive number at the beginning and just work out fractions of it on the way across?"

The Mathematical Ninja shuffled his feet. "You might think that," he said. "I couldn't possibly comment."

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. The Mathematical Ninja is brilliant at game theory []

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