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Day trading An option to buy a stock is priced at $$\$ 200$$. If the stock closes above 30 on May \(15,\) the option will be worth $$\$ 1000$$. If it closes below \(20,\) the option will be worth nothing, and if it closes between 20 and 30 (inclusively), the option will be worth $$\$ 200$$. A trader thinks there is a \(50 \%\) chance that the stock will close in the \(20-30\) range, a \(20 \%\) chance that it will close above 30 , and a \(30 \%\) chance that it will fall below 20 on May \(15 .\) a. Should she buy the stock option? b. How much does she expect to gain? c. What is the standard deviation of her gain?

You play two games against the same opponent. The probability you win the first game is 0.4 . If you win the first game, the probability you also win the second is 0.2 . If you lose the first game, the probability that you win the second is 0.3 . a. Are the two games independent? Explain. b. What's the probability you lose both games? c. What's the probability you win both games? d. Let random variable \(X\) be the number of games you win. Find the probability model for \(X\). e. What are the expected value and standard deviation?

1 is 0.8 . If you get contract #1, the probability you also get contract #2 will be 0.… # Your company bids for two contracts. You believe the probability you get contract #1 is 0.8 . If you get contract #1, the probability you also get contract #2 will be 0.2, and if you do not get #1, the probability you get #2 will be 0.3 a. Are the two contracts independent? Explain. b. Find the probability you get both contracts. c. Find the probability you get no contract. d. Let \(X\) be the number of contracts you get. Find the probability model for \(X\) e. Find the expected value and standard deviation.

In a group of 10 batteries, 3 are dead. You choose 2 batteries at random. a. Create a probability model for the number of good batteries you get. b. What's the expected number of good ones you get? c. What's the standard deviation?

In a litter of seven kittens, three are female. You pick two kittens at random. a. Create a probability model for the number of male kittens you get. b. What's the expected number of males? c. What's the standard deviation?

A grocery supplier believes that in a dozen eggs, the mean number of broken ones is 0.6 with a standard deviation of 0.5 eggs. You buy 3 dozen eggs without checking them. a. How many broken eggs do you expect to get? b. What's the standard deviation? c. What assumptions did you have to make about the eggs in order to answer this question?

A company selling vegetable seeds in packets of 20 estimates that the mean number of seeds that will actually grow is \(18,\) with a standard deviation of 1.2 seeds. You buy 5 different seed packets. a. How many bad (non-growing) seeds do you expect to get? b. What's the standard deviation? c. What assumptions did you make about the seeds? Do you think that assumption is warranted? Explain.

A delivery company's trucks occasionally get parking tickets, and based on past experience, the company plans that each truck will average 1.3 tickets a month, with a standard deviation of 0.7 tickets. a. If they have 18 trucks, what are the mean and standard deviation of the total number of parking tickets the company will have to pay this month? b. What assumption did you make in answering?

An insurance company estimates that it should make an annual profit of $$\$ 150$$ on each homeowner's policy written, with a standard deviation of $$\$ 6000$$. a. Why is the standard deviation so large? b. If it writes only two of these policies, what are the mean and standard deviation of the annual profit? c. If it writes 10,000 of these policies, what are the mean and standard deviation of the annual profit? d. Is the company likely to be profitable? Explain. e. What assumptions underlie your analysis? Can you think of circumstances under which those assumptions might be violated? Explain.

The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces. You open a new box of cereal and pour one large and one small bowl. a. How much more cereal do you expect to be in the large bowl? b. What's the standard deviation of this difference? c. If the difference follows a Normal model, what's the probability the small bowl contains more cereal than the large one? d. What are the mean and standard deviation of the total amount of cereal in the two bowls? e. If the total follows a Normal model, what's the probability you poured out more than 4.5 ounces of cereal in the two bowls together? f. The amount of cereal the manufacturer puts in the boxes is a random variable with a mean of 16.3 ounces and a standard deviation of 0.2 ounces. Find the expected amount of cereal left in the box and the standard deviation.