Activity 6 - 1

1. Activity 6 - 1 Chances Are! Gerolamo Cardano, Italian mathematician, wrote the first book about Probability in 1560.

2. Odds are ….

3. Objectives • Determine relative frequencies for a collection of data • Determine both theoretical and experimental probabilities • Simulate an experiment and observe the law of large numbers • Identify and understand the properties of probability

4. Vocabulary • Relative Frequency – what percentage of the whole is some interested part • Event – an occurrence of something, like rolling a six on a single die • Probability of an Event – the chances of an event occurring • Random – • Sample Space – • Probability Distribution – • Theoretical Probability – • Experimental Probability – • Simulation –

5. Activity The study of probability began with mathematical problems arising from games of chance. In 1560, Italian GerolamoCardano wrote a book about games of chance. This book is considered to be the first written on probability. Two French mathematicians, Blaise Pascal and Pierre de Fermat, are credited by many historians with the founding of probability theory. They exchanged ideas on probability theory in games of chance and worked together on the geometry of the die. Although probability is most often associated with games of chance, probability is used today in a wide range of areas, including insurance, opinion polls, elections, genetics, weather forecasting, medicine, and industrial quality control.

6. Activity cont Complete the following table (which lists years of service of the 789 workers at High Tech Manufacturing). What is the total number of males? What is the total number of workers with 1-10 years of service? 490 299 131 563 95 490 563

7. Activity cont Convert the table to a table of Relative Frequencies (Example: 82/789 = .104) If a person is randomly selected from the company, what is the probability that ….? • They are a male • They have less than one year experience • They are a woman with 1-10 years experience • They don’t work for the company 0.458 0.060 0.621 0.062 0.256 0.061 0.379 0.166 0.714 0.120 1.000 490/789 = 0.621 131/789 = 0.166 202/789 = 0.256 0/789 = 0

8. Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run The unpredictability of the short run entices people to gamble and the regular and predictable pattern in the long run makes casinos very profitable.

9. Randomness and Probability We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term frequency.

10. Relative Frequency • Relative frequency is the percentage that the observed makes up of the whole • Its found by dividing the number of a given category by the total number of values • It is equivalent to the Experimental Probability

11. Probability • Experimental Probability • Based on observed frequencies of events • Theoretical Probability • Based on theoretical frequency of events frequency of the event Probability of an event = ------------------------------------------- total number of observations number of outcomes of the event Probability of an event = --------------------------------------------------- total number of possible outcomes

12. Laws of Probability Let P(x) be the probability that event x occurs • Collection of all possible outcomes is called the sample space • 0 ≤ P(x) ≤ 1 for all events x in sample space • Sum of all P(x) for all events x must equal 1 • P( certainty ) = 1 • P( impossibility ) = 0

13. Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion (experimental probability) with which a certain outcome is observed get closer to the theoretical probability of the outcome.

14. Random Selection Laws of Probability depend on the supposition that all objects in the collection have an equal chance of being selected

15. Example 1 Using a six-sided dice, answer the following: a) P(rolling a six) b) P(rolling an even number) b) P(rolling 1 or 2) d) P(rolling an odd number) 1/6 3/6 or 1/2 2/6 or 1/3 3/6 or 1/2

16. Example 2 Identify the problems with each of the following a) P(A) = .35, P(B) = .40, and P(C) = .35 b) P(E) = .20, P(F) = .50, P(G) = .25 • P(A) = 1.2, P(B) = .20, and P(C) = .15 • P(A) = .25, P(B) = -.20, and P(C) = .95 ∑P > 1 ∑P < 1 P() > 1 P() < 0

17. Summary and Homework • Summary • Law of Large Numbers: as the number of trials is increased the experimental probability approaches theoretical probability • Properties of Probability: a) Sum of probabilities of all possible events must equal one b) Probability of any single event must be between 0 and 1 c) Probability of impossibility is zero d) Probability of certainty is one • Theoretical Probability = (number of outcomes in event) / (total number of all possible outcomes) • Experimental Probability = (number of observed occurrences of event) / (total number of observations) • Homework • pg 706 – 712; problems 1, 5, 7, 8, 9, 11