Assignment:
1. Suppose that you draw a single card from a standard deck of 52 playing cards.
a. What is the probability that this card is a diamond or club?
b. What is the probability that this card is not a 4?
c. Given that this card is a black card, what is the probability that it is a spade?
d. Let E1 be the event that this card is a black card. Let E2 be the event that this card is a spade. Are E1 and E2 independent events? Why or why not?
e. Let E3 be the event that this card is a heart. Let E4 be the event that this card is a 3. Are E3 and E4 independent events? Why or why not?
2. A fair coin (i.e., heads and tails are equally likely) is tossed three times. Let X be the number of heads observed in three tosses of this fair coin.
a. Find the probability distribution of X.
b. Find the probability that two or fewer heads are observed in three tosses.
c. Find the probability that at least one head is observed in three tosses.
d. Find the expected value of X. e. Find the standard deviation of X.
3. Three areas of southern California are prime can- didates for forest fires each dry season. You believe (based on historical evidence) that each of these areas, independently of the others, has a 30% chance of having a major forest fire in the next dry season.
a. Find the probability distribution of X, the number of the three regions that have major forest fires in the next dry season.
b. What is the probability that none of the areas will have a major forest fire?
c. What is the probability that all of them will have a major forest fire?
d. What is expected number of regions with major forest fires?
e. Each major forest fire is expected to cause $20 million in damage and other expenses. What is the expected amount of damage and other expenses in these three regions in the next dry season?
4. The probability distribution of X, the number of cus- tomers in line (including the one being served, if any) at a checkout counter in a department store, is given by P(X = 0) = 0.25, P(X = 1) = 0.25, P(X = 2) = 0.20, P(X = 3) = 0.20, and P(X = 4) = 0.10.
a. Use simulation to generate 500 values of this random variable X.
b. Is the simulated distribution indicative of the given probability distribution? Explain why or why not.
c. Calculate the mean and standard deviation of the simulated values. How do they compare to the mean and standard deviation of the given probability distribution?
5. An investor has invested in nine different investments. The dollar returns on the different investments are probabilistically independent, and each return follows a normal distribution with mean $50,000 and standard deviation $10,000.
a. There is a 1% chance that the total return on the nine investments is less than what value?
b. What is the probability that the investor's total return is between $400,000 and $520,000?
6. A production process manufactures items with weights that are normally distributed with mean 15 pounds and standard deviation 0.1 pound. An item is considered to be defective if its weight is less than 14.8 pounds or greater than 15.2 pounds. Suppose that these items are currently produced in batches of 1000 units.
a. Find the probability that at most 5% of the items in a given batch will be defective.
b. Find the probability that at least 90% of the items in a given batch will be acceptable.
c. How many items would have to be produced in a batch to guarantee that a batch consists of no more than 1% defective items?
7. Suppose you play a game at a casino where your prob- ability of winning each game is 0.49. On each game, you bet $10, which you either win or lose. Let P(n) be the probability that you are ahead by at least $50 after n games. Graph this probability versus n for n equal to multiples of 50 up to 1000. Discuss the behavior of this function and why it behaves as it does.
8. Suppose that X, the number of customers arriving each hour at the only checkout counter in a local pharmacy, is approximately Poisson distributed with an expected arrival rate of 20 customers per hour.
a. Find the probability that X is exactly 10.
b. Find the probability that X is at least 5.
c. Find the probability that X is no more than 25.
d. Find the probability that X is between 10 and 30 (inclusive).
e. Find the largest integer k such that we can be at least 95% sure that X will be greater than k.
f. Recalling the relationship between the Poisson and exponential distributions, find the probability that the time between two successive customer arrivals is more than four minutes. Find the probability that it is less than two minute.
9. In the decade 1982 through 1991, 10 employees work- ing at the Amoco Company chemical research center were stricken with brain tumors. The average employ- ment at the center was 2000 employees. Nationwide, the average incidence of brain tumors in a single year is 20 per 100,000 people. If the incidence of brain tumors at the Amoco chemical research center were the same as the nationwide incidence, what is the probability that at least 10 brain tumors would have been observed among Amoco workers during the decade 1982 through 1991? What do you conclude from your analysis? (Source: AP wire service report, March 12, 1994).