Theoretical and Subjective Probability

Theoretical Probability

What is Theoretical Probability?

For any sample of size n, the empirical probability that a randomly chosen individual from the sample has property E is given by the following formula: \[P(E)=\dfrac{n(E)}{{n}}=\dfrac{\text{number of the objects with property }E}{\text{sample size}}\]

Important Note

This is just our formula for relative frequency.  We will discuss the relationship between theoretical probability, empirical probability, and relatively frequency shortly.

Example 1

In a sample of 30 people, it was found that 18 had blue eyes. What is the probability that, when you select one of the 18 people at random, you get a person with blue eyes?

Solution

\[P(\text{{blue}})=\dfrac{\text{number with blue eyes}}{\text{sample size}}=\dfrac{{18}}{{30}}=\frac{{3}}{{5}}=0.6\]

$$\tag*{\(\blacksquare\)}$$

Example 2

Follow this link to open a StatCrunch simulation on flipping a fair quarter.

  • Part A: Flip a fair coin 10 times.  What proportion of heads did you get?
  • Part B: Flip a fair coin 10,000 times.  What proportion of heads did you get?

Solution

Since the results are random, watch the lecture video to see what was done in class.

$$\tag*{\(\blacksquare\)}$$

Borel's Law of Large Numbers

For a sufficiently large sample size \(n\), \[\text{empirical probability }\approx\text{theoretical probability}\]

Probability and Relative Frequency

What is the difference between probability and relative frequency?

  • Relative Frequency tells us what percentage of the data has a given property.
  • Empirical Probability estimates the percentage of time a property should appear when an experiment is repeatedly done forever.
  • Theoretical Probability is the exact percentage of time a property appears when an experiment is repeatedly done forever.

Subjective Probability

What is Subjective Probability?

The probability of an event E occurring is estimated by using knowledge of the relevant circumstances.

Example 3

Estimate the probability that the next time you turn on a light bulb, you discover that it is still in working order. Why isn't this a good way to determine probability?

Solution

Each person will come up with a different probability, and people are bad judges of how often theings occur.