Introduction to Probability Probabilities are expressed in terms of events Examples of Events 1 2 3 4 5 6 7 8 9 Six shows on a roll of a die Jack of Spades is drawn from a deck of cards Rains at 3 00 p m today in front of the Reitz Union Have an automobile accident this year Florida beat Florida State Florida beat Tennessee Florida beat Florida State and Tennessee Yield of a randomly drawn citrus tree is greater than 8 boxes Mean yield of 25 randomly drawn citrus trees is greater than 8 boxes Events are denoted by capital letters A B etc Probabilities of events are denoted P A or P event occurs Determination of Probabilities Probabilities are determined from 1 Relative frequency computation 2 Subjective assessment P six on roll of die 1 6 a relative frequency computation P Florida beat Florida State 3 a subjective assessment P Rain today at 3 00 at JWRU 4 combination of relative frequency and subjective assessment Compound Events and Probabilities of Compound Events Compound events are formed from combinations of events The union of events A and B occurs if either A or B occur Florida beat FSU U Florida beat Tennessee Florida beat either FSU or Tennessee Florida beat FSU Florida beat Tennessee Florida beat both FSU and Tennessee Calculating Probabilities of Compound Events P AUB P A P B P A B P Florida beat either FSU or Tennessee P Florida beat FSU P Florida beat Tennessee P Florida beat both FSU and Tennessee Two events A and B are independent if P A B P A P B Are the events Florida beat FSU and Florida beat Tennessee independent Are the events Drives 4WD truck and Voted Republican independent Are the events Lives in Florida and Voted Republican independent Conditional Probability The conditional probability of A given B is the probability that A occurs when it is known that B occurs It is calculated by the formula P A B P A B P B If A and B are independent then P A P A B Examples P J P draw J from deck of cards 4 1 52 13 13 1 P C P draw club from deck of cards 52 4 Intersection P J C P J and C Union 1 52 P J C P J or C P J P C P J C 1 1 1 4 13 1 16 13 4 52 52 52 P J Q 0 P J Q 8 2 52 13 Dependence P J C P J P C J and C are independent P J Q P J P Q J and C are not independent Application Prostate Cancer 50 1 005 10000 200 398 0398 P PSA 10000 48 P PC PSA 0048 10000 P PC PSA Test Conditional Probability P A given B P A B 48 350 398 2 9600 9602 50 9950 10000 P A B P B P PSA PC 48 50 96 sensitivity P PSA PC 9600 9950 9648 specificity P PC PSA 48 398 12 predictive ability Random Variables Discrete spots on top face of die suit of drawn card aphids on leaf defects in box of 1000 nails germinating seeds out of 50 Continuous heights of people ph of soil voltage in circuit 0 0 10 0 1 2 3 4 5 6 C D H S 0 1 2 3 0 1 2 1000 0 1 2 50 Binomial Random Variable Let x Number of successes out of n trials in which the probability of success on each trial is a number Then x has a binomial distribution with parameters n and This is abbreviated x B n Example Consider flipping a coin and declare a success if a head H appears Then the probability of success is 5 That is P S 5 Suppose the coin is flipped n 3 times and x number of heads Then x B 3 5 The possible values of x are 0 1 2 3 The probability of any event is 1 2 3 1 8 Events HHH HHT Heads 3 2 P Event 1 8 1 8 P 3 H 1 8 HTH 2 1 8 THH HTT THT TTH TTT 2 1 1 1 0 1 8 1 8 1 8 1 8 1 8 P 2 H 3 8 P 1 H 3 8 3 3 3 1 1 In general P k H k 3 k 2 k 3 k 8 0 1 2 3 P 0 H 1 8 Binomial Random Variable con t Example x number of 1 s in 3 rolls of die 0 1 2 3 Events 111 11X 1X1 X11 1XX X1X XX1 XXX 1 s 3 2 2 2 1 1 1 0 3 3 2 1 3 2 1 3 2 1 3 1 2 3 1 2 3 1 2 3 3 P Event 1 6 1 5 6 1 5 6 1 5 6 1 5 6 1 5 6 1 5 6 5 6 0 579 P 3 1 s 1 1 1 1 1 3 0046 6 6 6 6 216 k 3 1 5 P k 1 s k 3 k 6 6 0 347 0 069 3 k 0 005 0 1 Binomial Formula probability of success on single trial n y n y P y successes in n trials y n y 1 Mean of the binomial distribution n Variance of the binomial distribution n 1 2 3 Normal Distribution and normal random variable The notation y N 2 means The random variable y is distributed normally with mean and variance 2 The standard normal distribution has mean 0 and variance 2 1 The letter z is reserved to represent the standard normal random variable Computer programs and tables are available to obtain probabilities from the normal distribution For example you can discover that P 1 z 1 68 P z 1 16 P 1 96 z 1 96 95 P z 1 96 025 P z 1 42 0778 Using the Normal Distribution Standardizing a Normal Distribution If y N 2 then z y N 0 1 This result allows us to compute probabilities from any normal distribution using tables or a computer program for the standard normal distribution If you wanted to calculate the probability that a random variable y is greater that 1 42 standard deviations above its mean you would compute y P y 1 42 P 1 42 P z 1 42 0778 As a more specific application suppose you believe the egg weights to be normally distributed with mean 65 4 and standard deviation 5 17 You would calculate the probability that a randomly drawn egg is greater than 70 as P y 72 P y 65 4 5 2 72 65 4 5 2 P z 1 27 1 Using the normal distribution an application Egg weights are normally distributed with mean 65 g and standard deviation 5 0 1 What is the probability one randomly drawn egg will exceed a 65 b 66 c 70 d 75 Let y egg weight …
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