A random experiment gives rise to possible outcomes, but any particular outcome is uncertain – “random”. For example, tossing a coin… we know H or T will appear, but on any one toss it is uncertain as to which it will be. An event is one of the many possible outcomes arising from a random experiment. Probability is a numerical measure of the likelihood an event will occur. Its possible values range from 0 (“impossible event”) to 1 (“certain event”) – think of it as the long run relative frequency of the event’s occurrence… Equally likely outcomes are defined as: If a random experiment has k possible outcomes then they are equally likely if each has prob. 1/k

0 <= P(A) <= 1 for all events A Some definitions… The sample space (S) of a random experiment is the collection of all possible outcomes An event (usually represented by A or B or…) is an outcome or a collection of outcomes from a random experiment (i.e., a subset of S) Probabilities are then computed on events so that these important rules hold: 0 <= P(A) <= 1 for all events A P(S) = 1 where S is the sample space P(A does not occur) = 1 – P(A) P(A or B) = P(A) + P(B), if A and B have no outcomes in common (disjoint or mutually exclusive) P(A and B) = P(A) P(B), if A and B are independent.

Sample spaces can be either finite or infinite Toss a fair coin. S = {H, T}; Spin a fair penny. S = ? Toss a fair coin twice. S = {HH, HT, TH, TT} Shoot a free throw 3 times. S = { ... ? ... } Toss a fair coin n times. S = { … ? … } Roll a fair die (one of a pair of dice) S = ? Roll a pair of fair dice. S = ? Toss a fair coin until the first Head occurs. S = ? Pick a digit at random from Table B. S = ? Record the weight gain of an experimental animal after 4 weeks on a high-protein diet. S = ? We model the real-world by assigning probabilities to the elements of the sample space so the total probability is = 1.

HW: Get ready for the test on Wednesday Then begin reading the intro to Chapter 5 and section 5.1… we’ll continue this on Friday and all of next week.