PROBABILITY AND STATISTICS WEEK 3 Onur Doğan 2016-2017

Introduction to Probability The Classical Interpretation of Probability Theoretically The Experimental Interpretation of Probability Empirically The Subjective Interpretation of Probability Subjectively Onur Doğan 2016-2017

Sample Space, Experiment, Event An experiment is any process, real or hypothetical, in which the possible outcomes can be identified ahead of time. An event is a well-defined set of possible outcomes of the experiment. Onur Doğan 2016-2017

Recall: Operations of Set Theory Onur Doğan 2016-2017

Example Example: Consider tossing a fair coin. Define the event H as the occurrence of a head. What is the probability of the event H, P(H)? 1. In a single toss of the coin, there are two possible outcomes 2. Since the coin is fair, each outcome (side) should have an equally likely chance of occurring 3. Intuitively, P(H) = 1/2 (the expected relative frequency) Notes: This does not mean exactly one head will occur in every two tosses of the coin In the long run, the proportion of times that a head will occur is approximately 1/2 Onur Doğan 2016-2017

Experiment Experimental results of tossing a coin 10 times each trial: Onur Doğan 2016-2017

Law of Large Numbers Law of Large Numbers: If the number of times an experiment is repeated is increased, the ratio of the number of successful occurrences to the number of trials will tend to approach the theoretical probability of the outcome for an individual trial Interpretation: The law of large numbers says: the larger the number of experimental trials n, the closer the empirical probability P(A) is expected to be to the true probability P(A) In symbols: As

Probabilities Expected value = 1/2 Trial Onur Doğan 2016-2017

Axioms and Basic Theorems Onur Doğan 2016-2017

mutually exclusive / exhaustive Onur Doğan 2016-2017

Example Two dice are cast at the same time in an experiment. Define the sample space of the experiment. Find the pairs whose sum is 5 (A) and the pairs whose first die is odd (B). Are A and B mutually exclusive? Are A and B exhaustive? Onur Doğan 2016-2017

Example In a city, 60% of all households subscribe to the newspaper A, 80% subscribe newspaper B, and 10% of all households do not subscribe any newspaper. If a household is selected at random, What is the probability that it subscribes to at least one of the two newspapers? Exactly one of the two newspapers? Onur Doğan 2016-2017

Example a. What is the probability that the individual has a medium auto deductible and a high homeowner’s deductible? b. What is the probability that the individual has a low auto deductible? A low homeowner’s deductible? c. What is the probability that the individual is in the same category for both auto and homeowner’s deductibles? d. What is the probability that the two categories are different? e. What is the probability that the individual has at least one low deductible level? f. What is the probability that neither deductible level is low? Onur Doğan 2016-2017

Counting Techniques Onur Doğan 2016-2017

Permutations Suppose that four-letter words of lower case alphabetic characters are generated randomly with equally likely outcomes. (Assume that letters may appear repeatedly.) How many four-letter words are there in the sample space S? How many four-letter words are there are there in S that start with the letter ”k”? What is the probability of generating a four-letter word that starts with an ”k” ? Onur Doğan 2016-2017

Permutations How many words of length k can be formed from a set of n (distinct) characters, (where k ≤ n), when letters can be used at most once ? Onur Doğan 2016-2017

Permutations An ordered subset is called a permutation. The number of permutations of size k that can be formed from the n individuals or objects in a group will be denoted by P(n,k). Onur Doğan 2016-2017

Permutations Sampling with Replacement. Obtaining Different Numbers. Birthday Problem? Onur Doğan 2016-2017

Combination Consider a set with n elements. Each subset of size k chosen from this set is called a combination of n elements taken k at a time. Onur Doğan 2016-2017

Example Suppose that a club consists of 25 members and that a president and a secretary are to be chosen from the membership. We shall determine the total possible number of ways in which two people will fill the two positions. Onur Doğan 2016-2017

Example Suppose that a class contains 15 boys and 30 girls, and that 10 students are to be selected at random for a special assignment. We shall determine the probability exactly three boys will be selected. Onur Doğan 2016-2017

Example Suppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so that each player receives 13 cards. We shall determine the probability that each player will receive one ace. Onur Doğan 2016-2017

Example Suppose that a fair coin is to be tossed 7 times, and it is desired to determine; the probability p of obtaining exactly three heads the probability p of obtaining three or fewer heads. Onur Doğan 2016-2017

Example A box containing 3 red, 4 blue and 5 black balls. What’s the probability that drawn 3 balls will be different colours? Onur Doğan 2016-2017

Binomial Coefficients Onur Doğan 2016-2017

Multinomial Coefficients Onur Doğan 2016-2017

Multinomial Coefficients Onur Doğan 2016-2017

Example Onur Doğan 2016-2017

Example How many nonnegative integer solutions are there to x + y + z = 17 ? Onur Doğan 2016-2017

Example What is the probability the sum is 9 in three rolls of a die ? Onur Doğan 2016-2017