Basic Probability

Theoretical versus Empirical Theoretical probabilities are those that can be determined purely on formal or logical grounds, independent of prior experience. Empirical probabilities are estimates of the relative frequency of an event based by our past observational experience.

Theoretical Probability Probability of A tossed coin landing on heads Drawing a spade from a poker deck Observing a three when rolling a die

Empirical Probability Empirical probability can be subdivided into two categories: Objective versus Subjective Probability that conception will result in twins (Objective) Probability of an insurance applicant filing a claim (Objective)

Objective Probability The previous examples can be considered objective in the sense that they are based on observations of past occurrences of events, under what are hopefully the same conditions that currently prevail.

Subjective Probability Empirical in the sense that they are ultimately based on past observation Subjective in the sense that the particular observation(s) upon which the particular probability estimate(s) are based, is not well defined, that is, a independent observer could not be instructed on how to arrive at the same probability

Subjective Probability What is the probability that a space satellite will fall out of orbit and land on Tucson? What is the probability that a direct-response advertisement will draw a profitable response? What is the probability of extra-terrestrial life? What is the probability that upon graduation, you will be offered a position on your first job interview?

Basic Probability Concepts

Probability Experiments Whenever we manipulate or make an observation on our environment with an uncertain outcome, we have conducted an experiment. Examples Taking an exam Tossing a coin Delivering a sales pitch

Probability Experiment Can be repeated many times (at least in theory) Each such repetition is called a trial When an experiment is performed it can result in one or more outcomes, which are called events.

Sample Space The set of all possible outcomes of an experiment is called the sample space, S, for the experiment The outcomes in the sample space are called sample points The outcomes forming the sample space must be mutually exclusive and exhaustive The sample space and sample points depend on what the experimenter chooses to observe

Example – Toss a Coin Twice Can record the sequence of heads (H) and tails (T), then S= {HH, HT, TH, TT} Can record the total number of tails observed, then S= {0, 1, 2} Can record whether the two tosses match (M) or do not match (D), so S= {M, D}

Exercise (Sample Spaces) Determine the sample space of the following experiments: Toss a die and recording the number on the top face Let a light bulb burn until it burns out Observe General Electric common stock and recording whether it increased, decreased or remained unchanged during one market day Record the sex of successive children in a three- child family

Events An event, E, is a subset of the sample space and it denoted by  An event E is said to occur if the outcome of an experiment is an element of E Consider the experiment of tossing a die once and recording the number on the top face. The sample space, S= {1, 2, 3, 4, 5, 6}

Example (Events) Some events associated with this experiment are: E 1 ={1} We observe a 1 E 2 ={2} We observe a 2 E 3 ={1,3,5} We observe an odd number E 4 ={1,2,3} We observe a number less than 4.

Simple vs Compound Events A simple event cannot be decomposed. A compound event is an event that can be decomposed into other events. E 1 and E 2 are simple events. E 3 and E 4 are compound events.

Exercise Consider the experiment of flipping a balanced coin three times. Determine the sample space for the experiment List two events that correspond to this experiment

Teminology Experiment Sample space Sample points Probability model Events Certain event Impossible event Mutually exclusive (disjoint) events

Discrete Sample Space A discrete sample space consists of a finite number of sample points or a countable number of sample points. Throughout Project 1, we will be concerned with finite discrete sample spaces.

Probability of an Event Given an event, we would like to assign it a number, P ( E ), called the probability of E This number indicates the likelihood that the event will occur. We can find this number by determining the value of the ratio:

Relative Frequency Suppose that we repeat the die tossing experiment n times and notice that the event E 1 occurs f times. We call the ratio f / n the relative frequency of the event after n repetitions. If we repeat this experiment indefinitely and if the ratio f / n approaches a number, p, as n becomes larger and larger, then p is called the probability of the event.

Law of Large Numbers The more repetitions we take, the better the approximation p  f / n This is sometimes referred to as the Law of Large Numbers, which states that if an experiment is repeated a large number of times, the relative frequency of the outcome will tend to be close to the probability of the outcome.

Summary of 20,000 Coin Tosses Num of TossesNum. of HeadsRelative Freq. nff / n , ,0002, ,0005, ,0007, ,00010,

Fundamental Properties Upon analyzing the relative frequency concept, we see the following must hold: 1.Negative relative frequencies do not make sense 2.The relative frequency of the sample space must be 1 3.If two events are mutually exclusive, the relative frequency of their union must be the sum of their relative frequencies.

Fundamental Properties Cont. 1.For an event E, 0  P ( E )  1 2.P ( S )=1, where S is the sample space 3.P ( E  F )= P ( E )+ P ( F ), where E and F are mutually exclusive events 4.P ( E 1  E 2  …  E k ) = P ( E 1 )+ P ( E 2 )+…+ P ( E k ), where the E k ’s are mutually exclusive.

Calculating P ( E ) 1.Define the experiment and clearly determine how to describe one simple event 2.List the simple events associated with the experiment and test each to be certain that they cannot be decomposed. This defines the sample space S.

Calculating P ( E ) Continued. 3.Assign probabilities to the sample points in S making certain that the Fundamental Properties for a discrete sample space are preserved. 4.Define the event, E, as a specific collection of sample points. 5.Find P ( E ) by summing the probabilities of the sample points in E.

Example A balanced coin is tossed three times. Calculate the probability that exactly two of the three tosses results in heads.

Example A balanced coin is tossed three times. Let E 1 be the event that you observe at least two heads. What is P ( E 1 )? Let E 2 be the event that you observe at exactly two heads. What is P ( E 2 )? Let E 3 be the event that you observe at most most heads. What is P ( E 3 )? What can you say about E 1 and E 3

Basic Theorems of Probability Let S be a discrete sample space. Theorem 1: P (  )=0, where  is the empty set. Theorem 2: For any two events E and F in S, P ( E  F )= P ( E ) + P ( F ) - P ( E  F ) Theorem 3: If E is an event in S, then P ( E C )= 1 - P ( E )

Mutually Exclusive Two events are mutually exclusive if A  B= . If A and B are mutually exclusive, then AB

Mutually Exclusive If no two events E 1, E 2,..., E n can happen at the same time, then