Theoretical Probability Section 2 Two types of probabilities

Theoretical Probability Section 2

Two types of probabilities Empirical Probability Theoretical Probability Empirical probability tells us the frequency that an event happened during repeated trials of an experiment. Theoretical probability allows us to predict the frequency of those repeated trials.

Theoretical Probability We assume that each outcome of an experiment happens with equal chance. The theoretical probability of an event E occuring in an experiment is Throughout the course, we’ll refer to theoretical probability as just probability.

Example There are six different outcomes of the experiment “rolling a six-sided die”: 1, 2, 3, 4, 5 and 6. Let E be the event of rolling a 2. The probability of E is

All possible outcomes of “rolling a die” experiment The event of rolling a 2

Sample space The sample space of an experiment is the set of all of the outcomes of that experiment. For example, the sample space of the “rolling a die” experiment is the set {1, 2, 3, 4, 5, 6}. We can then think of events as subsets of the sample space.

Sample space Event E Sample space P(E) = # outcomes in E / # outcomes in sample space

Sample space of “roll a die” Event of rolling a 2 Sample space

More examples with “roll a die” B (empty set) C Events A = “a number greater than 4” P(A) = 2/6 = 1/3 B = “ 7” P(B) = 0/6 = 0 C = “a number less than 7” P(C) = 6/6 = 1 Sample space A

Complements The complement of an event E is the event within the sample space as E in which E does not occur. We denote it with EC. Also called “not E”. We have the following two identities between E and EC:

Complements Event EC Event E Sample space

Complements in “roll a die” sample space DCC (“a B (empty number set) less than or equal to 4”) AC Events A = “ 2” B = “a number greater than 4” D = “a number less than 7” Sample space

Question Suppose we flip a coin 10 times. Which of the following is more likely to happen? HHHTHHTTTH Or HHHHH Both are equally likely!

For the “rolling a six-sided die” experiment, what is the probability of rolling either an even number or rolling a number less than or equal to 3? “Even number” = {2, 4, 6} “Number less than or equal to 3” = {1, 2, 3} “Even number OR number less than or equal to 3” = {1, 2, 3, 4, 6} Probability = 5/6

Sample space of “roll a die” “Number less than or equal to 3” Sample space “Even number”

P(“even number OR number ≤ 3”) = P(“even number”) + P(“number ≤ 3”) - P(“even number AND number ≤ 3”) =½+½-⅙ =⅚ In general,