Section 6.2: Definition of Probability

Probability of an event E denoted P(E) is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space: This method is appropriate only when the outcome is equally likely

Example: Calling the Toss On some football teams, the honor of calling the toss at the beginning of a football game is determined by random selection. Suppose that this week a member of the offensive team will call the toss. There are 5 interior linemen on the 11-player offensive team. If we define the event L as the event that lineman is selected to call the toss, 5 of the 11 possible outcomes are included in L. The probability that a lineman will be selected is then P(L) = 5/11

Example: Math Contest Four students (Adam, Betty, Carlos, and Debra) submitted correct solutions to a math contest with two prizes. The contest rules specify that if more than two correct responses are submitted, the winners will be selected at random from those submitting correct responses. In this case, the set of possible outcomes for the chance experiment that consists of selecting the two winners from the four correct responses is: {(A,B), (A,C), (A,D), (B,C), (B,D), (C,D)}

Because the winners are selected at random, the six outcomes are equally likely and the probability of each individual outcome is 1/6. Let E be the event that both selected winners are the same gender. Then: E = {(A,C), (B,D)}

Because E contains two outcomes, P(E)=2/6 =.333. If F denotes the event that at least one of the selected winners is female, the F consists of all outcomes except (A,C) and P(F) = 5/6 =.833

Probability – Empirical Approach Law of large numbers – As the number of repetitions of a chance experiment increases, the chance that the relative frequency of occurrence for an event will differ from the true probability of the event by more than any small number approaches 0.

Relative Frequency Approach to Probability The probability of an event E, denoted by P(E) is defined to be the value approached by the relative frequency of occurrence of E in a very long series of trials of a chance experiment. Thus, if the number of trials is quite large,

Methods for Determining Probability 1.The classical approach: Appropriate for experiments that can be described with equally likely outcomes. 2.The subjective approach: Probabilities represent an individual’s judgment based on facts combined with personal evaluation of other information. 3.The Relative Frequency Approach: An estimate is based on an accumulation of experimental results. This estimate, usually derived empirically, presumes a replicable chance experiment.

Section 6.3: Basic Properties of Probability

Basic Properties of Probability For any event E, 0  P(E)  1. If S is the sample space for an experiment, P(S)=1. If two events E and F are disjoint, then P(E or F) = P(E) + P(F). For any event E, P(E) + P(not E) = 1 so, P(not E) = 1 – P(E) and P(E) = 1 – P(not E).

Property One: Suppose we are flipping a bottle cap to find the success of caps landing face up. Suppose that after N times, we have x successes. What are the possible values of x? The least amount it can be is 0. The most it could be is N. The relative frequency between them is 0 to 1.

Property Two: Because the probability of any event is the proportion of time an outcomes in the event will occur in the long run and because the sample space consists of all possible outcomes for a chance experiment, in the long run an outcome in S must occur 100% of the time. Thus P(S)=1

Example: Cash or Credit? Customers at a certain department store pay for purchases with either cash or one of four types of credit card. Store records, kept for a long period of time, indicate that 30% of all purchases involve cash, 25% are made with the store’s own credit card, 18% with MasterCard (MC), 15% with Visa (V), and the remaining 12% with American Express (AE). The following table displays the probabilities of the simple events for the chance experiment in which the mode of payment for a randomly selected transaction is observed:

Simple event O 1 (Cash) O 2 (Store) O 3 (MC) O 4 (V) O 5 (AE) Probability Let’s create event E, the event a randomly selected purchase is made with a nationally distributed credit card. This event consists of the outcomes MC, V, and AE. Therefore, P(E)=P(0 3 ∪ O 4 ∪ O 5 )=P(0 3 )+P(O 4 )+P(O 5 )= =.45 45% of all purchases are made using one of the three national cards.

In addition, P(not E) = 1 – P(E) = =.55 You could also say that not E consists of outcomes 1 and 2 with is =.55

Calculating Probabilities When Outcomes are Equally Likely Consider an experiment that can result in any one of N possible outcomes. Denote the corresponding simple events by O 1, O 2,… O n. If these simple events are equally likely to occur, then

Example: How likely is it you will be the mayor in Mafia? Consider the experiment consisting of randomly picking a card from an ordinary deck of playing cards (52 card deck). Let A stand for the event that the card chosen is a King.

Addition Rule for Disjoint Events Let E and F be two disjoint events. One of the basic properties of probability is P(E or F) = P(E ∪ F) = P(E) + P(F) This property of probability is known as the addition rule for disjoint events.

Example Consider the experiment consisting of rolling two fair dice and observing the sum of the up faces. Let E stand for the event that the sum is 7 and F stand for the event that the sum is 11.