Although winning the Texas Lottery Jackpot is a very unlikely event, there are still other prizes you can win. What is the Expected winnings on a $1 bet? Prize Amounts $20,000,000 (Matching 6#) $2,000 (Matching 5#) $50 (Matching 4#) $3 (Matching 3#) $ 0 (Lose) Odds 1 / 25,827,165 1 / 89,678 1 / 1,526 1 / 75 ? μx = $0.87 μx = $ -0.13

You are given the opportunity to draw TWO cards from a deck of cards You are given the opportunity to draw TWO cards from a deck of cards. You WIN $100 for every HEART you draw. Determine the Probability Model and determine how much money you would be willing to pay to play this game. Prize Amounts Odds $100 $200 $0

RULES FOR MEANS AND VARIANCES If you increase each random variable by a fixed number, b, or multiply by a scalar, a, what happens to the Mean and Standard Deviation? If you increase the grades by 1.2 points, what is the new Mean and Std. Dev.? If you multiply the grades by 1.2 points. What is the new Mean and Std. Dev.? μ(aX ± b) = a·μX ± b σ(aX ± b) = a·σX μx = 2.25 σx = 1.1779 Example 1: σx = 1.1779 μx = 3.45 μx = 2.70 σx = 1.4135

RULES FOR MEANS AND VARIANCES Combining Data Sets If you Add or Subtract two Independent Random Variables, X and Y, then the resulting Mean will be the Sum or Difference of the two means, but the new Variance is ALWAYS the Sum of the two variances. μ(X ±Y) = μX ± μY σ2(X ± Y) = σ2X + σ2Y Scaler: μ(AX ± BY) = AμX ± BμY σ2(AX ± BY) = A2σ2X + B2σ2Y NOTE: To compute all Standard Deviation questions, you must convert Std. Dev. to Variance, perform the operations and then square root result.

EXAMPLE: Company X produces a compact car with a mean gas mileage of 34 mpg and standard deviation 5.2. Company Y produces a compact car with gas mileage 25 and standard deviation 6.0. What is the Mean of the difference between the two automobile’s gas mileage? What is the Standard Deviation of the difference in gas mileage between two independent compact cars? μ(X ±Y) = μX ± μY σ2(X ± Y) = σ2X + σ2Y 1. μ(X - Y) = μX - μY μ(X – Y) = 34 – 25 = 9 mpg 2. σ2(X - Y) = σ2X + σ2Y σ2(X – Y) = 5.22 + 6.02 = 63.04 σ (X – Y) = √63.04 = 7.940 mpg

Let X be an independent random variable with mean 85 and standard deviation 16, and Let Z = 5X + 12. Find μZ and σZ: 5(85) + 12 = 437 μZ = E(Z) = 5·μx + 12 = σ2Z = b2·σ2X = 52 (162) = 6400 σ = 80 Let X have a mean of 85 and std. dev. of 16, and Let Y have a mean of 15 and std. dev. of 5, Find μ2x+y and σ2x+y: μ2x+y = 2·μx + μy = 2(85) + 15 = 185 σ22x+y = 22·σx2 + σy2 = 22(162) + 52 = 1049 σ2x+y =32.3882

Let X have a mean of 85 and std. dev Let X have a mean of 85 and std. dev. of 16, and Let Y have a mean of 15 and std. dev. of 5, Find μ3x+.5y+1 and σ 3x+.5y+1 : μ = 3·μx + .5μy + 1= 3(85) + .5(15) + 1 = 263.5 σ2 = 32·σx2 +.52 σy2 = 32(162) + .52(52) = 2310.25 σ =48.065

a.) How many broken eggs do you expect to get? COOL DOWN A grocery supplier believes that in a dozen eggs the mean number, X, of broken eggs is 0.8 with standard deviation 0.4 eggs. You buy 5 dozen eggs without checking. a.) How many broken eggs do you expect to get? b.) What is the standard deviation? c.) What assumptions did you have to make about the eggs in order to answer question (b)? μ(X+X+X+X+X) = (0.8)+ (0.8)+ (0.8)+ (0.8)+ (0.8) = 4 eggs VAR(X) = σ2(X+X+X+X+X) = (0.4)2+(0.4)2+(0.4)2+(0.4)2+(0.4)2 VAR(X) = σ2 = 0.80 σ = √0.8 = 0.8944 Each carton of eggs must be INDEPENDENT from the others.

HW: PAGE 383: 23-28

You are given the opportunity to draw ONE card from a deck of cards You are given the opportunity to draw ONE card from a deck of cards. You WIN $100 for drawing a HEART. BUT if you draw a 5, you win NOTHING. Determine the Probability Model and determine how much money you would be willing to pay to play this game. Prize Amounts Odds $100 $0