Saturday, October 10, 2009

      (questions )
  • Combinations, Permutations and Binomial
Probability and Statistics
  • Probability is a concept that is important in its own right; but also, it is the engine that drives the decisions of inferential statistics.
  • the principles that control probability are Randomness and chance variations .
  • Randomness means that you have no way of knowing which of several possible events will occur in a particular situation at a given time. The event that does occur is a random event and is due to chance.
  • Chance variations means that the event that did occur was not predictable ahead of time.
  • Probability is a number representing the likelihood that a particular event will occur by chance.
  • A random sample is a sample from a population in which all of the individual values in the sample were selected according to their likelihood or probability in the population.
  • Inferential statistics is the science of specifying the particular situations within which the observed events occur.
Probability: examples
  • Probability of Heads in one flip of a fair coin: p(H) = 1/2; p(T)=1/2=.5
  • Probability of getting a Heads and a Tails in 2 flips of a coin: p(H and T)=2/4=.5
  • Pr. of drawing a Spade from a standard well-shuffled deck of cards: p(S)=13/52=1/4=.25
  • p(3 of diamonds)=1/52= 0.019
  • p(roll a �7� in one roll of a pair of fair dice)=6/36=1/6=.167
  • Pr. of getting a woman in a single random selection from a class of 223 students with 150 women: p(w)=150/223=.673
  • Pr. of getting all women by random selection of 5 students from that class: p(5w)= 591,600,030/4,392,741,639 =.1347
  • Monte Hall Problem: Pr. of selecting the correct door as in the TV Show "Let's make a deal"
  • Probability is a number representing the likelihood that a certain kind of event will occur under a specified set of conditions . A probability is a number assigned to the likelihood of a possible single event.
  • Probabilities are numbers like proportions, that is, a number between 0 and 1. It is a number equal to the proportion of times an �A� event is likely to occur in N random chances of exactly the same kind
    • p(A)=1 means the event is almost certain to occur
    • p(A)=0 means the event is almost certain not to occur
Three definitions of Probability
  • A priori: rational analysis as in the examples.
  • Relative frequency: if there are many experiments of the same kind, the probability of a favorable event would be
Intuitive Probability : An evaluation of the probability based on informal criteria. In statistics all three definitions are used.

  • click for examples of probability spaces

    • Experiment: The conditions of generating a sample point. What an experiment is, is determined by the researcher.
    • Sample point or Simple (elementary) event (Hurlburt's "Outcome") [ = sample]: a single specific outcome of an experiment. A simple event generates a single point in a sample space. Note, if the experiment is flipping a coin 20 times, a simple event is HTTHHTHHHTTHTTTHHTHH.
    • Sample space: the set of all possible sample points which might occur in an experiment. A sample space may contain any number of sample points. There are 1048576 sample points in a space of 20 coin flips. Probabilities are defined over sample spaces.
    • Probability distributions are probabilities of certain events defined over sample spaces. [= sampling distributions]
    • An event: a subset of the sample space. It includes one or more sample points. Events may be defined in many ways. e.g. in 20 coin flips--heads on the first flip, three tails in a row; 10 heads; less than 3 heads. Each event has a probability.
    • Consider two events A and B defined over a single experiment
    • Intersection: The set of points that satisfy both A and B.
    • Union: The set of points that satisfy either A or B or both.
    • Independent events : Two events are independent iff the probability of B does not change whether or not A occurs and vice versa.
    • mutually exclusive events: Two events are mutually exclusive if A and B cannot occur in the same experiment.
    • partitioning the sample space : a set of mutually exclusive events which include every sample point
    • Multiplication law : If two events are independent
    • p(A and B) = p(A)* p(B) = p(A)p(B)
    • Addition law: p(A or B) = p(A) + p(B) - p(A and B)
    p(A or B) is often written p(A+B) p(A and B) is often written p(AB)
    Important formulas: Note the conditions under which they apply
    • For 2 mutually exclusive events: p(A+B) = p(A) + p(B)
    • For 2 mutually exclusive events: p(AB) = 0
    • The probability of the union of two independent events:
    • p(A+B) = p(A) + p(B) - p(A)p(B).
    • Conditional probability : The probability of getting an event A when you know event B has occurred p(A|B) = p(AB)/p(B)
    Iff A and B are independent p(A|B) = p(A) If A and B are mutually exclusive p(A|B) = 0
    p(A|B) does not necessarily equal p(B|A)
    • click for examples of probability spaces
    • Example: In a roll or a pair of dice, what is the probability of getting a total of 4 spots? What is the probability of getting a total of �4� if the first die is a �2�? If the first die is a �4�?

    Binomial Probability Distributions
    • Consider an experiment in which for each trial there are two possible outcomes, each trial can be considered independent of the previous ones, and the probabilities for each outcome considered constant
    • Examples: 1. flipping a coin N times; (Heads, Tails);
    • 2. asking N random people if they support President Clinton; (Y, N); 3. Guessing on an N item multiple choice test (right, wrong) 4. Number of one-spots in N rolls of a die (1, not one)
    • Such trials are called Bernoulli trials. An experiment consists of N such trials.
    • Events are classified by the number of "Hits" in the N trials. Each size N and each probability of a hit defines a different probability distribution
    Randomness and Probability
    is a binomial, each term in its expansion defines the probability of a random variable in its binomial probability distribution
    Binomial expansion: If a coin were flipped N, times the 1st term is
    P( 0 heads), 2nd term is P(1 head), rth term is P(r heads), Nth term is
    P (N heads).
    • Randomness and probability are co-defined. One can think that the particular event that occurs events in the sample space according to its probability. Example: You have a .5 chance to select H from a sample space containing H and T on a given trial if is a random trial and it has a probability of .5. You have about a .001 chance to get 10 heads in sampling from a sample space of 10 Bernoulli trials with a .5 probability of H.
    Binomial Distributions : continued
    • probability of a hit in a trial = p
    • probability of a miss = 1-p = q
    • N trials define an experiment. Each trial is independent of the others. A sample point is a sequence of N outcomes

    Binomial continued
    • If p=q=.5 Binomial is symmetrical
    • As N gets larger Binomial approaches a normal distribution. If N is 10 or more, the symmetrical binomial distribution is very similar to a normal distribution. Because of this fact one can often use the normal tables to estimate binomial probabilities. Also as N gets larger, the binomial becomes more symmetrical even when p does not equal .5.
    • For all binomial distributions, the Mean of the binomial  , (for proportions  )
    • Standard deviation , (for proportions  )
    In a binomial probability distribution the X-axis represents the number of Hits (in N trials) and the Y-axis is the probability of that X. The area under the curve represents the probability density. (for proportions, the X-axis represents the proportion of Hits (in N trials), the rest of the relationships are unaffected)
    If N is large the Normal curve is a good approximation to the Binomial.
    • Definition of Factorial: N! = N(N-1)(N-2)(N-3)�3*2*1, 0!=1, 1!=1
    • Consider a population that has N members and we generate a sample by selecting r individuals. The number of different samples there are in the population is the number of Combinations.
    • If they are selected one at a time the number of different sequences is the number of Permutations .
    •   Combinations
    • Permutations 

    Normal Curve
    The normal curve can be viewed as a probability distribution. The area under the curve equals one. The area between any two values on the abscissa is a number between zero and one. The probability of randomly selecting an individual with a score between those two values is equal to that area.Thus if you know the mean and standard deviation of a normal distribution, for example, an IQ test with m = 100, s = 16, you can figure out the probability of selecting an individual whose IQ is between 120 and 130. Or selecting someone whose IQ is less than 70. Problems:
    1. In a room with 200 students, how many different groups of 2 people can there be? How many of 5 people? How many of 20 people?
    2. What is the probability that someone selected at random from a population (m = 500, s = 100) would score 400 or less? What is the probability that three randomly selected people would all score 400 or less?

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