From Mathematical to Intuitive Definition of Probability

• Anyhow, the variable point X is now called a random variable and is the probability function of X.

• To relate this definition to the outside world, we must resort to the vanishing chalk dust approach as espoused by Mathews and Walker.

• If we flip an (unbiased) coin N times, we expect

  

  where is the number of times it turns up heads.

• We define

  

• This is a random variable with only two possible values --- heads or tails. Alternately, we could say it takes values x=0 and 1.

• Then, in our previous language any Borel set containing 0 and not 1 has , any containing 0 and 1, , etc.

• This illustrates the relation of intuitive frequency definition of probability and that based on mathematical measure.