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1. Relative Freq.: Conduct procedure and count the number of times that it occurs. 2. Subjective Prob.: P(A) is estimated by using knowledge of relevant circumstances. 3. Classical Approach: P(A) = (# ways A occur)/(# simple events)Probability Rules 0 ≤ P(x) ≤ 1 and P(x) = 0 (impossible) P(x) = 1 (certain) Unusual Event P(A) ≤ 0.05Law of Large Number: As a procedure is repeated many times, the relative freq. prob. tends to approach the actual prob. Odds Actual odds against event A happening P(A)/(P(A) a:b

Actual odds in favor of event A happening P(A)/(P(A) Pay-off odd against A = not profit : amount bet Addition Rule Mutually Exclusive P(A or B) = P(A)+P(B)

Not Mutually Exclusive P(A or B) = P(A)-P(B)Multiplication Rule Independent Event (Two events are independent if when A occurs, the prob B occurs is not affected) If the sample size is no more than 5% of the population, treat the selections as independent eventsP(A and B) = P(A)*P(B)

Dependent Event: P(A and B) = P(A)*P(B\A) (P(B\A) Conditional Probability is the probability that B occurs given A has occurred P(B\A)=P(A and B)/P(A))Complements P(at least x) = 1-P(at most x-1)Permutation (order matters) 1. There are n different items available 2. Select r of the n items r<n 3. Rearrangement of the same items to be different sequences

nPr= n!(n−r ) !

Combination (order doesn’t matter) 1. There are n different items available 2. Select r of the n items r<n 3. Rearrangement of the same items to be same sequences

nCr= n!(n−r ) !r !

Arrangement

¿of arrangement= n!n1 !n2 !…nk !

Probability Distribution (requirements: 1) P(x) = 1 and 2) 0 ≤ x ≤ 1) ΣUnusually high # if: P(x or more) ≤ 0.05 or +2 μ σ Unusually low # if: P(x or less) ≤ 0.05 or +2μ σBinomial Prob Distribution (requirements: 1) Procedure has a fixed # of trials n; 2) Trials are independent [sampling with replacement or sample ≤ 5% of pop.); 3) Each trial must have two categories; and 4) The prob of success remains the same)P(exactly x=__) = binomPdf(); P(at least x=__) = 1-P(at most x-1) = 1-binomCdf(); P(at most x=__) = binomCdf();Density Curve – Continuous Random Variable (requirements: 1) The area under the curve equals 1; 2) Density curve sits on or above x-axis. Vertical height of zero or above.)Uniform Distribution Probability – Graph is a rectangleStandart Normal Distribution – Graph is bell-shaped1. Sketch the normal curve, and label and ; 2. Label x values and shade the area desired; 3. Use the calculator to find area μ σ(probability)The functions used in the calculator is normalCdf(…) and invNorm(…)

The Central Limit Theorem: μx=μ and σ x=σ /√n1) If the original population is normally distributed, then for any sample size, the sample means are normally distributed2) If the original population is not normally distributed, then for a sample n>30, the sample means are normally distributedNormal as Approximation to Binomial (requirement: 1) Sample is a SRS; 2) np ≥ 5; 3) nq ≥ 5)If r is left point of an interval x = r-0.5 >>> P(r ≤ __ ) = P(x ≤ __-0.5) = normalCdf(…)If r is right point of an interval x = r+0.5 >>> P(r ≥ __ ) = P(x ≥ __+0.5) = normalCdf(…)

Ex) 1000 tickets are sold at $2 each for a prize valued at $650. What is the expected value of gain if a person purchases one ticket?

GAIN (x) P(x)Win 650-2 1/1000Lose -2 999/1000

E=648(1/1000)+(-2)(999/1000)=-1.35 The person expected to lose $-1.35 each time