# statistics

# Probability

### Conditional probability

I have three coins in my pocket. On one of them, both sides are "heads"; the second coin has two "tails" and the last coin is normal and has "heads" on one side, "tails" on the other.

I take out one of the coins at random and place it on a table. It shows heads.

What's the probability that the other side is also heads?

Have a think about this question before reading on. The actual answer may surprise you, and gives a useful introduction to the idea of conditional probability.

The Desmos page below simulates this experiment.

Run it at least fifty times (you need to click run and show other side for each trial).

Do you want to revise your original answer?

Combine your results with those of others in your class. Do you want to revise your original answer now?

Have a look at this Colab simulation, which repeats the experiment many times. Are you still convinced by your original answer?

Explain!

# Random variables

### King's random walk on a chessboard

A random variable to investigate. If a king starts at the bottom left square of the chessboard and takes random moves until he returns to his starting square, let X be the number of moves taken. Use the applet to investigate the distribution of X.

# Hypothesis testing

### A Hypothesis test using M&Ms

A Colab notebook introduction to hypothesis tests. The main focus is on the idea of testing a hypothesis, based on a seemingly non-uniform distribution of colours in a bag of M&Ms. The chi-squared goodness of fit test is used, but I have used this approach even with classes who don't need it as part of their specification. The Desmos applet below accompanies the sheet - press get new M&Ms packet to generate a random treat bag!

### Normal distribution test of mean

A Colab notebook discussing how to test the sampling mean of a normal distribution.

# Normal distribution

### Probability density function

A structured set of questions aimed at deriving the probability density function of the normal distribution, together with a Desmos graph to illustrate the approach to question 4, and a nice print to card version too.

# Poisson distribution

### Poisson from Binomial

A derivation of the Poisson distribution from the binomial.