Sandsynlighed Note

Intuitive Understanding of Probability: Many people have an intuitive sense of probabilities, but formal definitions can be elusive.
Example: The barber's lottery strategy illustrates a primitive understanding of probability: "Either I win, or I lose, so I have a fifty-fifty chance."
Everyday Probabilities: We often consider probabilities in daily life, such as the chance of flipping heads, someone getting married, or rain during a game.
Confusion: Defining probability precisely can be confusing.
Expressing Belief: Probabilities reflect a degree of belief in situations where outcomes are uncertain.
Varying Certainty: While the probability of flipping heads might be seen as 50-50, other probabilities are harder to define without specific information.
Learning Objectives: Readers should be able to:
- Differentiate and use three types of probability concepts.
- Determine simple probabilities within these concepts, using computers.
- Argue for the correctness of additive law and the law of complementary events.
- Understand the mathematical formalization of probability with probability spaces and random variables.

Daily Language: We often use everyday language to express our degree of belief.
- Examples: "I can't possibly make the meeting at 10" or "It's possible I'll make the meeting at 10."
Uncertainty: Even strong statements like "I will definitely make the meeting" indicate some uncertainty.
Qualitative Statements: These statements should not always be taken precisely, as they express a lack of certainty.
Rules for Qualitative Probabilities:
- If A is more likely than B, then B is less likely than A.
- If A is more likely than B, and B is more likely than C, then A is more likely than C.
Exercise 1: Consider other terms for qualitative steps on the probability scale beyond "impossible", "fifty-fifty" and "certain".
Exercise 2: Rank the following events by likelihood:
- Having at least three children.
- Becoming a school principal.
- Passing the teacher exam in mathematics on the first try.
- Living to be at least 70 years old.
- Celebrating a silver wedding anniversary.
Percentages: Probabilities are often expressed as percentages, where 100% is total certainty and 0% is impossible.
Exercise 3: Relate qualitative terms from Exercise 1 to a percentage interval between 0 and 100.

Children's Understanding: There can be a problem when understanding children's more intuitive perceptions of probabilities corresponding to terms like maybe, possible, impossible, 50-50, and certain.
Research Findings: Children do not always use these terms as expected. Research indicates that children generally use the expressions consistently with normal language usage.
Peculiarities: Some peculiarities occur for some of the children.
Example: A fourth-grade student says it is impossible to get the number 31 in a tombola with numbers between 1 and 20, while a fifth-grade student says it is certain to get 31 because there is a 31 in the tombola.
50-50: Also, children use 50-50 even when there is not a 50% chance for each of the two possible outcomes.
Danish Studies: There are no Danish studies of children’s use of everyday language in chance situations. One can expect differences between users of different languages.
Conclusion: It is expected the Danish children sometimes use that kind of word differently than expected.

Three Quantitative Probability Concepts:
- Subjective
- Statistical
- Combinatorial
Objective: To define what they have in common and formulate a core quantitative probability concept.

Definition: Simplest and most unclear form of probability.
Example: If person A says he is 100% sure to come at 10 am, that is true regardless of whether he comes or not.
Subjective probability: It does not need to be defined from the point of view of others.
Provocative Language: It can be provocative to say, "It is 200% certain that I will come tomorrow at 10 am", but from a probability point of view, it does meet the general requirement that the degree of conviction is quantified in the interval [0;1].
Bookmakers: Subjective probabilities are determined by bookmakers. For example, a person is willing to bet 1:20 that Denmark will win the next World Cup in football.
Combining It: By combining subjective probability with the statistical probability concept, it is possible to interpret the subjective probability as a frequency.
Rational Action: A person's action balances as a rational action at the considered subjective probability if the person experiences the excitement to have an extra bonus.
Exercise 4
- Bookmaker's perspective to determine a person's probability of Klaus Rifbjerg receiving the Nobel Prize in Literature within the next four years.

Experiment: A repeated action or event.
Different Outcomes: An experiment can have different outcomes.
Statistical Probability: Repeat the experiment many times and note how often the given outcome occurs.
Formula:
$P(udfaldet) = \frac{Antal \, gange \, udfaldet \, forekommer}{Totale \, antal \, gange \, eksperimentet \, udførelse}$
Problems: Defining