Counting Techniques, Probability, Normal Distribution, Mathematical ExpectationsProbability Distribution

What is Probability Distribution?Aprobability distributionis a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. Consider the coin flip experiment described above. The table below, which associates each outcome with its probability, is an example of a probability distribution.

An example will make clear the relationship between random variables and probability distributions. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.

Number of headsProbability00.2510.5020.25

Uniform Probability DistributionThe simplest probability distribution occurs when all of the values of a random variable occur with equal probability. This probability distribution is called theuniform distribution. Uniform Distribution.Suppose the random variable X can assume k different values. Suppose also that the P(X = xk) is constant. Then,P(X = xk) = 1/k

Example 1 Suppose a die is tossed. What is the probability that the die will land on 6 ?Solution:When a die is tossed, there are 6 possible outcomes represented by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is a random variable (X), and each outcome is equally likely to occur. Thus, we have a uniform distribution. Therefore, the P(X = 6) = 1/6.

Example 2 Suppose we repeat the dice tossing experiment described in Example 1. This time, we ask what is the probability that the die will land on a number that is smaller than 5 ?Solution:When a die is tossed, there are 6 possible outcomes represented by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is equally likely to occur. Thus, we have a uniform distribution.This problem involves a cumulative probability. The probability that the die will land on a number smaller than 5 is equal to:P( X < 5 ) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1/6 + 1/6 + 1/6 + 1/6 = 2/3

COUNTING TECHNIQUESPermutation and Combination

Fundamental Counting PrincipleThe Fundamental Counting Principle(FCP)If one thing can occur m ways and a second thing can occur in n ways, and a third thing can occur in p ways, and so on, then the sequence of things can occur inm x n x p x ways

Example 1:The Shirt mart sells shirts in sizes S, M, L and XL.. Each size comes in five colors: red, yellow, white, orange, and blue. The shirts come in short sleeve and long sleeve. How many kinds of shirts are there?

PermutationFACTORIAL NOTATIONIn General, if n is a positive integer, then n factorial denoted by n! is the product of all integers less than or equal to n.n! = n.(n-1).(n-2). .2.1As special case, we define 0! = 1

Definition:A permutation is the ordered arrangement of distinguishable objects without allowing repetitions among the objects.The number of permutations of n things taken n at a time is given bynPn = n(n-1)(n-2) (3)(2)(1) = n!The permutation of n things taken r at a time is given bynPn = n(n-1)(n-2) (n-r+1)

Example 1:In how many ways can a president and vice-president be chosen from a club with 12 members?

If there are 50 floats in Penagbenga Festival, how many ways can a first-place, a second-place, and a third-place trophy be awarded?

Example 2:Find the number of different arrangements of the set of six letters HONESTa. Taken two at a timeb.Taken three at a timec. Taken six at a time.

Example 3:Find the number of permutations in each situation.a. A softball coach chooses the first, second, and third batters for a team of 10 players.b. Three-digit numbers are formed from the digits 2,3,4 and 5, with no digits repeated.

Permutation of Identical ObjectsThere are instances when the n things to be arranged are not all different. That is r1 of them are alike, r2 of them are alike, rk of them are alike where r1+r2+ rk =n. The number of permutations of these things therefore is

Practice Exercises:1. In how many different ways can four people be seated in a row?2.In how many ways may the letters of the word STATISTICS be arranged?3. In how many ways may the letters of the word ASSESSMENT be arranged?

Circular PermutationIf n different things are to have a circular arrangement then the number of Permutations is equal; to

(n-1)

Example 1:

In how many ways can 3 keys be arranged in a key ring? Combination

The number of combinations of n things taken r at a time is given by

Example:Mr. Elton has to choose three of the six officers of the MATH Club to go to a regional meeting. How many possible choices does he have?

How many Combinations of 5 records can be chosen from 12 records offered by a record club?