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A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability 0.3, and his second will lead independently to a sale with probability 0.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. Determine the probability mass function of X, the total dollar value of all sales. Figure out the sample space first S={(no sale,no sale), (no sale, deluxe), (no sale, standard) (deluxe, no sale) (standard, no sale) (deluxe standard) , (standard, deluxe) (deluxe, deluxe), (standard, standard) } Then figure out the probabilities and the values of X X =0, 500, 1000, 1500, 2000 To get, for example, the probability of 500, that is P{(no sale, standard), (standard, no sale)} = (prob no sale in the first times the probability of making a sale in the second and this sale is for standard) + (probability of making a sale in the first and this sale is for standard times the probability of not making a sale in the second) = (0.7x0.5x0.4)+ (0.5x0.3x0.4) Write that in a table format. 2.
Four independent flips of a fair coin are made. Let X denote the number of heads obtained. Plot the probability mass function of the random variable X
2. 1 (not graded) 9.
If the cumulative distribution function of X is given by Calculate the probability mass function of X. p(0) = 1/2 (not graded
not assigned) p(1) = 3/5
1/2 = 1/10 p(2) = 4/5
3/5 = 1/5 p(3) = 9/10
4/5 = 1/10 p(3.5) = 1
9/10 = 1/10 2 3.
A gambling book recommends the following “winning strategy” for the game of roulette. It recommends that a gambler bet 1 on red. If red appears (which has probability 18/38), then the gambler should take her 1 profit and quit. If the gambler loses this bet (which has probability 20/38 of occurring), she should make additional 1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote the gambler’s winnings when she quits. (a) Find P(X > 0) (hint: write first the probability mass function of X). (b) Are you convinced that the strategy is indeed a “winning” strategy? Explain your answer. (c) Find E(X). Let represent the outcome by 1 if winning $1 and 0 if losing $1. Outcomes Net gain 1 011 010 001 000 (a) P(X>0)= 18/38+20/38*18/38*18/38 = 0.5918 (b) No. Although P(X>0)>0.5, when X<0, X can be –3, which is a bigger loss. We need to find the expectation to be sure. 3 1 1
1
1
3 Probability 18/38 20/38*18/38*18/38 20/38*18/38*20/38 20/38*20/38*18/38 20/38*20/38*20/38 (c) P(X=
1)= 20/38*18/38*20/38*2=0.2624 P(X=
3)=0.1458 E(X)= 0.5918
0.2624
3*0.1458=
0.108 This means it is not a ‘winning strategy”. 4.
If E(X) = 1 and Var(X) = 5, find (a) E[(2+X)2] (b) Var(4+3X) E(X)=1 Var(X)=E(X2)
[E(X)]2=5 E (X2)=6 (a) E[(2+X)2]=E[4+4X+X2]=4+4E(X)+E(X2)=4+4+6=14 (b) Var(4+3X)=Var(3X)=9Var(X)=9*5=45 7. The manager of a stock room in a factory knows from his study of records that the daily demand (number of times used) for a certain tool has the following probability distribution: Quantity demanded Probability 0 0.1 1 0.5 2 0.4 (In other words, 50 % of the daily records show that the tool was use one time). Let X denote the daily demand. (a) How much can would he expect the tool to be used tomorrow? (b) By how much could he be off? (c) Suppose that it costs the factory $10 each time the tool is used. What is the expected daily cost of using the tool? How much is this cost expected to vary from one day to another? Done in class. This is in the course reader lesson 6 8. It is known that diskettes produced by a certain company will be defective with probability 0.01, independently of each other. The company sells the diskettes in packages of size 10 and offers a moneyback guarantee that at most 1 of the 10 diskettes in the package will be defective. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them? 4 The three packages are independent of each other. Find the return probability for each of them first: 7.
The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be (a) at least 2 such accidents in the next month? (b) at most 1 accident in the next month? 8..
The number of times that a person contracts a cold in a give year is a Poisson random variable with parameter λ=5. Suppose that a new wonder drug (based on large quantitites of vitamin C) has just been marketed that reduces the Poisson parameter to λ=3 for 75% of the population. For the other 25 percent of the population the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her? 5 6 ...
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This document was uploaded on 03/20/2011.
 Winter '09
 Probability

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