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You are playing a game of yahtzee you throw dice and must roll exactly sixes to win what

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y aces or twos, you lose the game immediately. You also lose if you draw picture cards(J,Q,K) more than twice. In this question, you’ll study the probability of winning this game.(a) What is the probability of drawing no aces or twos after thirteen draws?(b) Given you have drawn thirteen times, none of which is aces or twos, what is the probability that you draw at most two picture cards?(c) What is the probability to win this game? 12. Suppose you are tossing an unbiased coin for100times.(a) What is the probability of getting50heads and50tails?(b) LetXbe the random variable counting the number of heads you observe in this exper-iment. What is the expected value ofX? What is the variance ofX? What is thestandard deviation ofX? 13. The following are probability distributions for two random variablesX,Y. kPr(X=k) 0,0.4 1,0.3 2,0.3 kPr(Y=k) 0,0.5 1,0.3 2,0.2 (a) Construct the probability distribution table for the random variableXY.(b) Find E[X],E[Y] and E[XY]. Is is true that E[XY] =E[X]E[Y]?(c) Find the variances σ2X,σ2Y,σ2XY of X,Y and XY. Is it true that σ2XY=σ2Xσ2Y? 14. The aliens who are fond of gambling came back to play another game with you. In this game, you first toss a coin5times. If you observe3or fewer tails, you roll a die3times. If youobserve4or more tails, you roll a die20times. What is the probability that you end up with at most two6’s in your dice rolls? 15. (Challenge question, worth2points) You have two bags, each of which contains10marbles.Each time you remove a marble from a random bag. What is the probability that after one of the bags is emptied, there are still exactly3marbles in the other bag?
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ards, where a face card is a Jack, a Queen, or a King, in succession from a standard deck of 52 cards without replacing the first card, you win \$10. Otherwise, you pay your friend \$20. If the same bet was made 15 times, how much would you expect to win or lose? Round your answer to the nearest cent, if necessary.
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e probability of this occurring?
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with a mean of 25 gm and a standard deviation of 5 gm. (a) If the machine is used 500 times, approximately how many times will it be expected to dispense 30 gm or more of chilli sauce? (b) How can you decrease this number to half? Give a numerical answer. 2. StarTech manufactures re sensors. They use a protective screen for their sensors to protect it from dust. The sensor becomes useless if the thickness of the screen exceeds 0.5 mm. They outsource the production of the screen to a di erent company that claims to manufacture screens with a mean thickness of 0.3 mm and a standard deviation of 0.1 mm. (a) If 10000 screens are manufactured how many will be discarded because they are too thick? (b) If screens less than 0.2 mm are too thin to be used, what is the probability that screens manufactured by the above company will be discarded because they are too thick or too thin? Show the result on a graph. 3. The amount of time that Sam spends playing the guitar is normally distributed with a mean of 15 hours and a standard deviation of 3 hours. (a) Find the probability that he spends between 15 and 18 hours playing the guitar during a given week. (b) What is the probability that he spends less than 3 hours playing the guitar during a given week? 4. Soon after he took oce in 1963, President Johnson was approved by 160 out of a sample of 200 Americans. With growing disillusionment over his Vietnam policy, by 1968 he was approved by only 70 out of a sample of 200 Americans. (a) What is the 90% con dence interval for the percentage of all Americans who approved of Johnson in 1963? In 1968? (b) What is the 90% con dence interval for the change?
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1.AU MAT 120 Systems of Linear Equations and Inequalities Discussion

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