PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 6 Conditional Probability 

Question Suppose that a certain precinct contains 350 voters, of which 250 voters are Democrats and 100 are Republicans. If 30 voters are chosen at random from the precinct, what is the probability that exactly 18 Democrats will be selected?

Question If A, B, and D are three events such that what is the value of

Question If the letters s, s, s, t, t, t, i, i, a, c, are arranged in a random order, what is the probability that they will spell the word “statistics”?

Conditional Probability The updated probability of event A after we learn that event B has occurred is the conditional probability of A given B. Conditional probability of A given that B has occurred.

Example Find the probability of a 1, given the occurrence of an odd number, in the toss of a single die.

Independence Two events, A and B, are said to be independent , if Otherwise, the events are said to be dependent.

Independence - Disjoint events Two events, A and B, are said to be independent , if Two events are disjoint if

Example Consider the following two events in the toss of single die. A: observe an odd number B:observe an even number C:observe a 1 or 2. Are A and B independent events? Are A and C independent events?

Example Three brands of coffee, X, Y and Z, are to be ranked according to taste by a judge. Define the following events: A: brand X is preferred to Y, B: brand X is ranked best, C: brand X is ranked second best D: brand X is ranked third best. If the judge actually has no taste preference and thus randomly assigns ranks to the brands, is event A independent of events B, C, and D?

Two laws of Probability Additive law: The probability of union of two events A and B is If A and B are mutually exclusive events,

Two laws of Probability Multiplicative law: The probability of the intersection of two events A and B is If A and B are independent,

Two laws of Probability Multiplicative law: Prove the probability of the intersection of any number of events

Bayes’ Rule Suppose that the events form a partition of the space and Then, for every event A in S,

Example Suppose that a person plays a game in which his score must be one of the 50 numbers 1, 2, …., 50 and each of these 50 numbers is equally likely to be his score. The first time he plays the game, his score is X. He then continues to play the game until he obtains another score Y such that Y≥X. Assume that all plays of the game are independent. Determine the probability of the event A that Y=50.

Example An electronic fuse is produced by five production lines in a manufacturing operation. The fuse are costly, are quite reliable, and are shipped to suppliers in 100-unit lots. Because testing is destructive, most buyers of the fuses test only a small number of fuses before deciding to accept or reject lots of incoming fuses. All five production lines normally produce only 2% defective fuses, which are randomly dispersed in the output. Unfortunately, production line 1 suffered mechanical difficulty 5 % defectives during the month of March. This situation became known to the manufacturer after the fuses had been shipped. A customer received a lot produced in March and tested three fuses. One failed. What is the probability that the lot was produced on line 1? What is the probability that the lot came from one of the four other lines?

RANDOM VARIABLE

Random Variable A real-valued function that is defined on the space S is called a random variable. The probability that X takes on the value of x, Pr(X=x), is defined to be the sum of the probabilities of all sample points in S which are assigned the value x by the function X. EX: Tossing a coin: A coin is tossed 10 times. Let X be the number of heads that are obtained.

Probability distribution The set of all pairs for which is called the probability distribution for X. EX: A foreman in manufacturing plant has three men and three women working for him. He wants to choose two workers for a special job. Not wishing to show any biases in his selection, he decides to select the two workers at random. Let X denote the number of women in his selection and find the probability distribution for X (as histogram)

Binomial distribution A binomial experiment is one that possesses the following properties: 1. The experiement consists of n identical trials. 2. Each trial results in one of two outcomes. For lack of a better nomenclature, we call one outcome a success, S, and the other a failure, F. 3. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to 1-p=q. 4. The trials are independent. 5. The random variable of interest is X, the number of successes observed during the n trials.

Binomial distribution

Binomial distribution Ex1: A coin is tossed ten times. Probability of observing seven times 1 or 2 Ex2: Consider a family having six children. Probability of observing two boys