Notes From David Palay : Chapter 5.1 Introduction to Probability

1. Notes From David Palay:Chapter 5.1Introduction to Probability What are the chances that…

2. Probability • From the book, • “The probability of an outcome is defined as the long-term proportion of times the outcome occurs.” • From Wikipedia, • “Probability is a way of expressing knowledge or belief that an event will occur or has occurred.” • Mr. David Palay, • “Probability is the chance something will or will not happen”

3. Terms • Experiment • An activity where the outcome is uncertain • NOT NECESSARILY UNKNOWN, JUST UNCERTAIN • Outcome • Result of a single trial of an experiment • Sample Space • Collection of all possible outcomes of an experiment • Event • Collection of outcomes from the sample space of an experiment

4. Rules of Probability • We write the probability of an event E as • Which means that the probability of any event is between 0 and 1. • 0 means it will NEVER EVER EVEREVER HAPPEN. • 1 means it will ALWAYS happen

5. Are these valid probabilities?

6. Rules of Probability (continued) • For any given experiment, the probability of the sum of the outcome probabilities in the sample space must equal 1. • SOMETHING has to happen, or we have an incomplete sample space.

7. Experiment & Theory • Experimental Probability: • Also called the “relative frequency method” • Probability we get from the results of running tests. • Theoretical Probability: • Also called the “classical method” • The probability calculated based on the rules of mathematical probability. (Which we will touch on later)

8. Dice nomenclature d – read “x dee y”, represents throwing x fair dice, each with y sides. e.g., • 3d6 (“three deesix”) represents rolling 3 six sided dice. • 1d20: 1 twenty sided die

9. Some Examples • Rolling a six: {6} • Rolling an even number: {2, 4, 6} • Rolling under a 3: {1,2} • Getting 2 heads {HH} • Getting at least 1 head {HH, HT, TH} • Picking a solid: {1, 2, 3, 4, 5, 6, 7, 8} • Picking a yellow ball {1, 9} • Picking the 8-ball {8}

10. Basic Probability

11. Ok, that sounds easy.. Find: P(rolling a 3 on 1d6): P(rolling odds on 1d6): Which is greater? Why?

12. More Practice • Standard deck of cards: 4 suits {Spades, Diamonds, Hearts, Clubs} and 2-10, Ace, Jack, Queen, King. The Jack, Queen, and King are considered “Face cards” P(drawing a 3 from a shuffled deck): P(drawing a face-card of hearts):

13. Slightly harder now… • What single sum has the highest probability of coming up when we roll 2d6? • We need to figure out how many possibilities there are. • Ah HA! Counting! We have 2 “spots”, each with 6 possibilities. So….

14. 2d6 continued

15. So, we can see…

16. Law of Large Numbers Given a sufficiently large number (infinite) of trials, the Experimental Probability will approach the Theoretical Probability

17. The Great Glass Rod Problem • If we take a glass rod, and break it at two random points, what is the probability that we will be able to form a triangle with the pieces.

18. Subjective Probability • Intuition. Guessing. Personal Judgement.