1. Section 12.1 The Nature of Probability

2. What You Will Learn History The Nature of Probability
Empirical Probability
The Law of Large Numbers

3. History
Probability is used in many areas, including public finance, medicine, insurance, elections, manufacturing, educational tests and measurements, genetics, weather forecasting, investments, opinion polls, the natural sciences, and games of chance.

4. History
Mathematical problems relating to games of chance were studied by a number of mathematicians of the Renaissance:
Italy’s Girolamo Cardano (1501–1576)
France’s Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665)
Dutch Christian Huygens (1629–1695)
Swiss Jacob Bernoulli (1654–1705)
Pierre-Simon de Laplace (1749–1827)

5. Definitions
An experiment is a controlled operation that yields a set of results.
The possible results of an experiment are called its outcomes.
An event is a subcollection of the outcomes of an experiment.

6. Definitions
Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment.
Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment.

7. Empirical Probability (Relative Frequency
The empirical probability of an event, P(E), can be determined by the following formula.

8. Example 1: Heads Up!
In 100 tosses of a fair coin, 44 landed heads up. Determine the empirical probability of the coin landing heads up.
Solution
Let E be the event that the coin lands heads up.

9. Empirical Probability in Genetics
Using empirical probability, Gregor Mendel (1822–1884) developed the laws of heredity by crossbreeding different types of “pure” pea plants and observing the relative frequencies of the resulting offspring. These laws became the foundation for the study of genetics.

10. Empirical Probability in Genetics

11. Empirical Probability in Genetics
Mendel called traits such as yellow color and round seeds dominant because they overcame or “dominated” the other trait. He labeled the green color and the wrinkled traits recessive.

12. Empirical Probability in Genetics

13. Empirical Probability in Genetics

14. Empirical Probability in Genetics
Mendel concluded that the sex cells (now called gametes) of the pure yellow (dominant) pea plant carried some factor that caused the off-spring to be yellow and that the gametes of the green variety had a variant factor that “induced the development of green plants.” In 1909, Danish geneticist W. Johannsen called these factors “genes.”

15. The Law of Large Numbers
The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals.

16. Tossing a Fair Coin