Review Homework Chapter 6: 1, 2, 3, 4, 13 Chapter 7 - 2, 5, 11 Probability Control charts for attributes Week 13 Assignment Read Chapter 10: “Reliability” Homework Chapter 8: 5, 9,10, 20, 26, 33, 34 Chapter 9: 9, 23 Week 12 Agenda
Probability Probability Chapter Eight
Probability Probability theorems Probability is expressed as a number between 0 and 1 Sum of the probabilities of the events of a situation equals 1 If P(A) is the probability that an event will occur, then the probability the event will not occur is P(A)
Probability Probability theorems For mutually exclusive events, the probability that either event A or event B will occur is the the sum of their respective probabilities. When events A and B are not mutually exclusive events, the probability that either event A or event B will occur is P(A or B or both) = P(A) + P(B) - P(both)
Probability Probability theorems If A and B are dependent events, the probability that both A and B will occur is P(A and B) = P(A) x P(B|A) If A and B are independent events, then the probability that both A and B will occur is P(A and B) = P(A) x P(B)
Probability Permutations and combinations A permutation is the number of arrangements that n objects can have when r of them are used. When the order in which the items are used is not important, the number of possibilities can be calculated by using the formula for a combination.
Probability Discrete probability distributions Hypergeometric - random samples from small lot sizes. Population must be finite samples must be taken randomly without replacement Binomial - categorizes “success” and “failure” trials Poisson - quantifies the count of discrete events.
Probability Continuous probability distributions Normal Uniform Exponential Chi Square F student t
Probability Fundamental concepts Probability = occurrences/trials 0 < P < 1 The sum of the simple probabilities for all possible outcomes must equal 1 Complementary rule - P(A) + P(A’) = 1
Probability Addition rule P(A + B) = P(A) + P(B) - P(A and B) If mutually exclusive; just P(A) + P(B) P(A)P(B) P(AandB)
Probability Addition rule example P(A + B) = P(A) + P(B) - P(A and B) Roll one die Probability of even and divisible by 1.5? Sample space {1,2,3,4,5,6} Event A - Even {2,4,6} Event B - Divisible by 1.5 {3,6} Event A and B ? Solution?
Probability Conditional probability rule P(A|B) = P(A and B) / P(B) A die is thrown and the result is known to be an even number. What is the probability that this number is divisible by 1.5? P(/1.5|Even)=P(/1.5 and even)/P(even) 1/6 / 3/6 = 1/3
Probability Compound or joint probability The probability of the simultaneous occurrence of two or more events is called the compound probability or, synonymously, the joint probability. Mutually exclusive events cannot be independent unless one of them is zero.
Probability Multiplication for independent events P(A and B) = P(A) x P(B) P(ace and heart) = P(ace) x P(heart) 1/13 x 1/4 = 1/52
Probability Computing conditional probabilities P(A|B) = P(A and B)/P(B) P(Own and Less than 2 years)?
Probability P(A)P(B) P(AandB) Computing conditional probabilities P(A|B) = P(A and B)/P(B)
Probability
Conditional probability Satisfied Not Satisfied Totals New Used Total S=satisfied N= bought new car P(N|S) = ?
Probability Just for fun 60 business students from a large university are surveyed with the following results: 19 read Business Week 18 read WSJ 50 read Fortune 13 read BW and WSJ 11 read WSJ and Fortune 13 read BW and Fortune 9 read all three How many read none? How many read only Fortune? How many read BW, the WSJ, but not Fortune? Hint: Try a Venn diagram.
Probability Probability Distributions
Probability Learning objectives Know the difference between discrete and continuous random variables. Provide examples of discrete and continuous probability distributions. Calculate expected values and variances. Use the normal distribution table.
Probability Random variables A random variable is a numerical quantity whose value is determined by chance. “A random variable assigns a number to every possible outcome or event in an experiment”. For non-numerical outcomes such as a coin flip you must assign a random variable that associates each outcome with a unique real number.
Probability Random variable types Discrete random variable - assumes a limited set of values; non-continuous, generally countable number of Mark McGwire homeruns in a season number of auto parts passing assembly-line inspection GRE exam scores
Probability Random variable types Continuous random variable - random variable with an infinite set of values. Can occur anywhere on a continuous number scale Baseball player’s batting average
Probability Random variables and probability distributions The relationship between a random variable’s values and their probabilities is summarized by its probability distribution.
Probability Probability distribution Whether continuous or discrete, the probability distribution provides a probability for each possible value of a random variable, and follows these rules: The events are mutually exclusive The individual probability values are between 0 and 1. The total value of the probability values sum to 1
Probability Probability distribution for rates of return Possible rate of return 10% 11% 12% 13% 14% 15% 16% 17% Probability Total = 1.0
Probability Describing distributions Measures of central tendency expected value (weighted average) Measures of variability variance standard deviation
Probability Expected value of a discrete random variable For discrete random variables, the expected value is the sum of all the possible outcomes times the probability that they occur. E(X) = {x i * P(x i )}
Probability Example: A fair die Roll 1 die: x P(x) x*P(x) E(x)=? 1 1/6 1/6 2 1/6 2/6 3 1/6 3/6 4 1/6 4/6 5 1/6 5/6 6 1/6 6/6 Can you sketch the distribution?
Probability Fair die illustrates a discrete “uniform distribution” The random variable, x, has n possible outcomes and each outcome is equally likely. Thus, x is distributed uniform.
Probability x P(x) 1/ Probability distribution
Probability Example: An unfair die Roll 1 die: x P(x) x*P(x) E(x)=? 1 1/12 1/12 2 2/12 4/12 3 2/12 6/12 4 2/12 8/12 5 2/12 10/12 6 3/12 18/12 Can you sketch the distribution?
Probability Expected value of a bet Suppose I offer you the following wager: You roll 1 die. If the result is even, I pay you $2.00. Otherwise you pay me $1.00. E(your winnings)=.5 ($2.00) +.5 (-1.00) = = $0.50
Probability Expected Value of a Bet Suppose I offer you the following wager: You roll 1 die. If the result is 5 or 6 I pay you $3.00. Otherwise you pay me $2.00. What is your expected value?
Probability Variance of a discrete random variable The variance of a random variable is a measure of dispersion calculated by squaring the differences between the expected value and each random variable and multiplying by its associated probability. {(x i -E(x)) 2 * P(x i )}
Probability Roll 1 die: [x- E(X)] 2 P(x) *P(x) / / / / /6.25 1/ /6.25 1/ / / / / Example: A fair die
Probability Probability distributions for continuous random variables A continuous mathematical function describes the probability distribution. It’s called the probability density function and designated ƒ(x) Some well know continuous probability density functions: Normal Beta Exponential Student t
Probability Continuous probability density function - Uniform If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by ab 1 b-a
Probability Uniform ab 1 b-a What is the probability of x? x
Probability Uniform ab 1 b-a Area under the rectangle = base*height = (b-a)* 1 = 1 b-a
Probability Uniform ab 1 b-a c P(c<x<b) = Area of brown rectangle 1 * (b-c) Ht x Width) = b-a
Probability Uniform P(2<x<5) = Brown rectangle 1 * (5-2) =(1/4) *3 = 3/4 = 5-1 = 1/4
Probability Uniform distribution If a random variable, x, is distributed uniform over the interval [a,b], then its pdf is given by And, the mean and variance are (a+b) ( b-a ) 2 E(x) = Var(x)=
Probability Uniform 38 Mean? Variance?
Probability And, the mean and variance are (a+b) ( b-a ) 2 25 E(x) = = 5.5 V(x)= = = So, if a = 3 and b = 8 Calculate uniform mean, variance
Probability Continuous pdf - Normal If x is a normally distributed variable, then is the pdf for x. The expected value is and the variance is 2.
Probability One standard deviation 68.3%
Probability Two standard deviations 95.5% 22 22
Probability Three standard deviations 99.73% 33 33
Probability Continuous PDF - Standard Normal If z is distributed standard normal, then and
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