Photo by Mike Hindle on Unsplash

A Machine Probability Puzzle

Will it ever work?

Here’s a simple puzzle to test your ability to think about probability.

Imagine you have a machine that works on average every 2 days out of 3. What is the probability that it will work for at least 4 of the next 5 days?

The solution is below. Take your time to solve this one! If you peek early, you hurt no one but yourself. Once you look, the problem is gone forever.

If you’re stuck, try making a diagram of all of the possible outcomes and calculating the probability of each of them.

The obvious first answer would be to multiply (2/3)⁴ by (1/3)¹ to get the chance that it works (which has probability 2/3) for 4 days and that it doesn’t work (which has probability 1/3) for 1 day. This gives me an answer of 16/243. However, this is only part of the story. There are 5 different ways this can occur, with the machine not working on each of the 5 days, so we need to multiply this fraction by 5.

I get a value of 80/243, which is almost the correct answer. We also need to add the probability that the machine works for all five days, which is given by (2/3)⁵ = 32/243.

Adding these probabilities together gives me an answer of 112/243.

This problem was inspired by a similar puzzle from the book Challenging Math Problems by Terry Stickels. It’s a fantastic source of puzzles like this one.

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