4-4 Counting 189 b. If 14 subjects are selected from the 16 that are available, and the 14 selected subjects are all treated at the same time, how many different treatment groups are possible? c. If 14 subjects are randomly selected and treated at the same time, what is the probability of selecting the 14 youngest subjects? 29. Mega Millions As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers from 1 to 70 and, in a separate drawing, you must also select the correct single number from 1 to 25. a. Find the probability of winning the jackpot. b. How does the result compare to the probability of being struck by lightning in a year, which the National Weather Service estimates to be 1>960,000? c. How does the probability compare to the probability for the old Mega Millions game which involved the selection of five different numbers between 1 and 75 and a separate single number between 1 and 15? 30. Medicare Identifiers In 2019, the Social Security Numbers used on Medicare cards were replaced with new Medicare Benefit Identifiers (MBI). Here is the format for these new MBIs: • Each MBI consists of 11 characters. • The character in position 1 will be a digit 1 through 9. • The characters in positions 4, 7, 10, 11 will be digits 0 through 9. • The characters in positions 2, 5, 8, and 9 will be uppercase letters, but the letters S, L, O, I, B, Z will not be used. The characters in positions 3 and 6 can be digits or letters (excluding S, L, O, I, B, Z). How many different MBIs are possible? Given that the current population of the United States is around 325,000,000, is the number of different possible MBIs sufficient? How does the number of different possible MBIs compare to the number of different possible Social Security Numbers used for Medicare cards before 2019? What is a major advantage of the new MBIs? 31. Morse Codes The International Morse code is a way of transmitting coded text by using sequences of on>off tones. Each character is 1 or 2 or 3 or 4 or 5 segments long, and each segment is either a dot or a dash. For example, the letter G is transmitted as two dashes followed by a dot, as in — — • . How many different characters are possible with this scheme? Are there enough characters for the alphabet and numbers? 32. Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type. 33. Change for a Quarter How many different ways can you make change for a quarter? (Different arrangements of the same coins are not counted separately.) 34. Counting with Fingers How many different ways can you touch two or more fingers to each other on one hand? 35. Design Your Own Lottery You have been given the task of creating a new lottery. For each $1 ticket, the player will select 6 different numbers from 1 to 25 (without replacement), and the only prize will be the jackpot won by players who select the six numbers (in any order) that are later drawn. a. What is the probability of winning with one ticket? b. What should be the winning prize if you want to average a profit of 50%, which is common for lotteries? c. Would this lottery have much appeal?
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