Introduction to Probability Day 11

Three Types of Probability I. Subjective Probability. This is probability based on ones’, (possibly educated), beliefs. Example. Your sister says that there is a 90% chance that she will be the lead in the school play next year. Example. The weather forecast is 20% chance of rain.

II. Empirical Probability. This is probability which is experimentally determined. Example. The following survey was given to 100 GCC sophomores: 1. (Yes or No.) Do you consider your experience in math classes at GCC an overall positive experience? 2. (Yes or No.) Have you taken more than one math class at GCC? Find the empirical probability that a GCC math student has (a) taken more than one math class at GCC; (b) had a negative experience in math classes at GCC; (c) has taken more than one math class or had a positive experience; (d) has taken more than one math class and had a negative experience. Survey Results Positive Experience Negative Experience Taken more than one2615 Taken no more than one2039

Three Types of Probability III. Theoretical Probability. This is probability which is based on mathematical theories and formulas. Example. A jar contains 80 white marbles, 100 blue marbles, 92 red marbles, 70 yellow marbles, and 85 green marbles. One marble is drawn from the jar at random. Find the theoretical probability that the marble is (a) red. (b) red or green. (c) red and green. (d) neither red nor green.

Definitions. (Single die example done in class with this.) A probability experiment is a controlled operation that yields a set of results. The results of a probability experiment are called outcomes. Each time an experiment is performed it is called a trial. The set of possible outcomes of an experiment is called the sample space. The sample space is usually denoted by the capital letter S. An event is a subset of the sample space.

An event containing exactly one element is called a simple event. An event containing more than one element is called a compound event. The event of the empty set is called an impossible event. The event of the sample space S is called a sure event. If all simple events in a probability experiment are equally likely, and E is a given event, then the following probability formula applies: P(E) = n(E)/n(S)

Properties of Probability 1. For all events E ⊆ S,, where P( φ) = 0 and P(S) = If is a list of all simple events in a sample space S, then

Example. A fair coin is tossed two times. (a) Give the sample space S in roster form. S = {HH, HT, TH, TT} (b) Let E be the event of two heads. List E in roster form. E={HH} (c) Let F be the event of at least one head. List F in roster form. F = {HH, HT, TH} (d) Find P(E) and P(F), (the probabilities of E and F). P(E) = n(E)/n(S) = ¼ =.25 = 25% P(F) = n(F)/n(S) = ¾ =.75 = 75%

Example: The cards below are shuffled and one card is drawn at random. (a) List the sample space and the event of a red card in roster form. (b) List the event of an even card in roster form.

Red CardsBlack Cards HeartDiamondClubSpade Ace (A,H)(A,D)(A,C)(A,S) Face Cards King (K,H)(K,D)(K,C)(K,S) Queen (Q,H)(Q,D)(Q,C)(Q,S) Jack (J,H)(J,D)(J,C)(J,S) 10 (10,H)(10,D)(10,C)(10,S) 9 (9,H)(9,D)(9,C)(9,S) 8 (8,H)(8,D)(8,C)(8,S) 7 (7,H)(7,D)(7,C)(7,S) 6 (6,H)(6,D)(6,C)(6,S) 5 (5,H)(5,D)(5,C)(5,S) 4 (4,H)(4,D)(4,C)(4,S) 3 (3,H)(3,D)(3,C)(3,S) 2 (2,H)(2,D)(2,C)(2,S)

(1,1)(1,2)(1,3)(1,4)(1,5)(1,6) (2,1)(2,2)(2,3)(2,4)(2,5)(2,6) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) (6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

Law of Large Numbers The Law of Large Numbers is a proven probability theory that states that if an experiment is performed many many times, then the empirical probabilities will be super close to the theoretical probabilities on average.

More Problems Monty Hall Problem