IE241 Problems 1. Three fair coins are tossed in the air. What is the probability of getting 3 heads when they land? 2. A town has 5000 adults and a random sample of 100 are asked for their opinion about a proposed sports arena. 60 of the respondents voted for it and 40 against it. If in fact the townspeople were half in favor and half against, what is probability that the sample would show 60% in favor? 3. If a box contains 40 good fuses and 10 defective fuses, and 10 fuses are chosen from the box at random. What is the probability that all 10 are good? 4. A die has two of its sides painted red, two painted yellow, and two painted black? If the die is rolled twice, what is the probability of two reds? 5. Assume that the ratio of male children is ½. In a family where 6 children are desired, what is the probability (a) that all 6 children will be of the same sex, and (b) that exactly 3 will be boys and 3 girls? 6. Compare the chances of rolling a 4 with 1 die and rolling a total of 8 with 2 dice. 7. The probability = ½ that a finesse in bridge will be successful. What is the probability that 3 out of 5 finesses will be successful? 8. A card is drawn from an ordinary deck. (a) What is the probability that it is a king, given that it is a face card (K,Q,J)? (b) What is the probability that it is a black king, given that it is a face card? 9. Two balls are drawn from an urn containing 2 white balls, 3 black balls, and 4 green balls. What is the probability that the first is white and second is black? 10. Consider a deck of cards consisting of only A, K, Q, J of each of the four suits. Find (a) the marginal distribution of suit and (b) the marginal distribution of face cards. 11. A company manufactures transistors in three different plants A, B, C, which use very similar procedures. It is decided to inspect the transistors in plant A because it is the largest. In order to test a week’s production, 100 transistors are selected at random and tested for defects. If 2 defects are found, what exactly can the company conclude about its transistor manufacturing operation? 12. A test of the breaking strength in pounds of 3/16” manila rope in a sample of 100 ropes produced an estimated mean of 550 and an estimated standard deviation of 80. Find a 95% confidence interval for the mean, assuming that the distribution of breaking strength is normal. 13. Compare the average length of a 95% confidence interval for the mean of a normal population based on the t distribution with the length of the confidence interval if σ were known. 14. Show that the length of the confidence interval based on t will approach 0 as n increases without bound. 15. A professor believes in the principle of a sound mind in a sound body, and decides to prove his point by measuring male students in his class on their running speed. He can’t measure the entire class so he decides to take the first 10 male students who show up for class on Monday. If he does this, how can he generalize his conclusions? 16. The diameters in feet of 56 shrubs were measured with the following results, where X = diameter and freq(X) = the frequency with which the value x occurred. X 1 2 3 4 5 6 7 8 9 12 freq (x) 1 7 11 16 8 4 5 2 1 1 Draw the histogram and calculate the mean diameter, the median diameter, and the mode. 17. Draw the histogram for the sum of the face numbers that come up when rolling two honest dice. What is the mean, median, and mode of this distribution? 18. Given that a binomial distribution has mean = 12 and variance = 8, find its parameters. 19. If one point on a binomial cdf is (3, 0.17), what is the probability of a value ≥ 4 ? 20. If one point on a normal cdf = (1.2, 0.37), what is the probability of a value ≥ 1.2 ? 21. Suppose you have two sets of data for the random variable X. The first set consisted of n1 observations and had mean X1 and standard deviation s1 and the second set consisted of n2 observations with mean X 2 and standard deviation s2. What is the mean of the combined group? 22. Suppose the November weather records show on average 3 out of 30 days of rain. Consider each day of November as an independent trial and compute the probability of at most 2 rainy days next November. 23. In the baseball world series, a team must win 4 games to win the series. The odds are 2:1 that team A will win each game. (a) What is the probability that team A wins in 4 straight games? (b) What is the probability that the series ends in 4 games? (c) What is probability that team A wins the series in 7 games? 24. A man takes 20 shots at a target. His probability of hitting the target is 1/10. What is the probability of at least 2 hits? 25. Give an example of two random variables for which the variance of their sum is (a) larger than the sum of their variances and (b) smaller than the sum of their variances. 26. Past experience indicates that wire rods purchased from company A have a mean breaking strength of 400 pounds and a standard deviation of 15 pounds. If 16 rods are selected, between what two values could you reasonably expect their mean to be? 27. An urn contains 2 red balls, 3 green balls, and 4 black balls. If 4 balls are drawn with replacement, what is the probability of getting 1 red, 1 green, and 2 black balls? 28. Three dice are thrown simultaneously. What is the probability of getting a total of 6 on the three dice? 29. Explain why it would not be surprising to find a correlation between traffic on Wall Street and tide height in Maine if you observed every hour from 6 am to 10 pm and high tide occurred at 8 am.