Bayes’ Formula Made Easy

Scientists often use Bayes’ formula when they test their scientific theories, but the formula doesn’t just apply in the realm of science. Bayes’ formula can also be useful in thinking about many situations encountered in daily life. Even if you can only make educated guesses about the probabilities involved, Bayes’ formula can improve the accuracy and consistency of those guesses. We’ll look at one of the simplest ways to use Bayes formula. Continue reading

Belief and Expectation

Your beliefs are your mental model of the world. When you see something happen in the world, your beliefs can often give you a way to explain it. You see water droplets turn sunlight into a rainbow. But having a good mental model of the world isn’t just about being able to explain things. Just as important – perhaps more important – are the things your beliefs can’t explain. In other words, your beliefs should narrow down the list of possible things you expect to observe. Continue reading

Thinking in Decibans

Probability is a way of thinking about chance that people use often in their daily lives. Probabilities are always between 0% and 100%, with something that is impossible having a probability of 0%, and something that is certain having a probability of 100%. Just like we can choose Kelvin, Celsius, or Fahrenheit scales to measure temperature, we can choose other scales for measuring chance besides probability, and these can be easier to think about in some situations. One such alternative measure of chance called log-odds has some very useful properties. Continue reading

Tall Giraffes and Hot Coffee

In science education the concept of a physical quantity is so basic that it is rarely defined explicitly, and students’ understanding of the concept is usually taken for granted. The even more basic concept of a physical quality is discussed more rarely still, but it is an interesting exercise to unpack and explore these concepts to gain a deeper understanding of what are, after all, some of the foundational ideas in science. Continue reading

How To Be Annoying and Unproductive

There is a misconception about science that seems to be common among non-scientists. That misconception is that science is about interpreting the world, about putting words together in a way that helps to provide a psychological frame for understanding the phenomena of the world. This is definitively not what science is about.

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The Cauchy Spinner Game

Imagine you are working at the local fairground midway and you’ve been assigned to operate an unusual carnival game called the “Cauchy Spinner”. The game is a spin the wheel type game; you spin a wheel with an arrow on it, and you win the amount that the arrow points to. The unusual thing about this game is that instead of the amount you win being arranged around the perimeter of the wheel, the payouts are shown as a long row of signs on the ground that seem to lead off forever into the distance. When the wheel stops, the amount a player wins depends on the distance to the spot on the ground where the arrow is pointing. How much do you have to charge people for each spin to make sure that you will make money in the long term? Continue reading

Three Ingredients

Imagine that I claim to have invented a new test for diagnosing some disease. Unlike many other tests, my test never gives false positives. In statistics we say the test is 100% specific; if you don’t have the disease, my test always correctly says you do not have the disease. This may seem like quite a remarkable achievement. Assuming my claims are true, would you rely on this test to diagnose you? Continue reading

Probability and Scientific Values

Science is often described as having a set of values or rules that guide the scientist in their quest for knowledge. Through history the great figures of science, from Aristotle to Bacon, Galileo and Descartes, to Einstein, Feynman and Sagan, have championed these values. These are used to justify the methods and reasoning that scientists use to understand the world, but this leads to the question: what is the justification for these values themselves? Continue reading