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Statistics Problems with answers

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Includes: permutation, combination, and probability problems. It also has detailed solutions and answers for these problems. Useful in quarter exams and Aptitude Tests. (Note: These questions are REALLY REALLY VERY HARD)
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  Questions (1) A fruit bowl has 6 apples and 4 oranges. If one piece of fruit is selected at random and then a second piece of fruit is selected at random, what are the chances that both pieces of fruit will be apples? (2) Golden Hoof (GH) is twice as likely to win a horse race than Silver Saddle (SS) and Silver Saddle is twice as likely to win as Chomping-at-the-Bit (CB). If only the 3 horses are in the race, what is the probability that Golden Hoof will not win? (3) Given a hand containing the nine numbered hearts from a deck of cards [2 to 9, with Ace counting as one (1)] — what is the probability of drawing three cards in sequence from the hand such that each selection results in a number larger than the previous one? (4) What are the odds against rolling a seven with a single throw of a pair of dice [2 dice]? (5) There are six men and twelve women. Half of the men have gray hair and so do half of the women. What are the chances that a person chosen at random will be a man, a person with gray hair or both? (6) How many dice must be thrown to give a better than 50% chance that at least one of the dice will show a one (1) (a single snake eye )? (7) If the chances of a golfer being struck by lightning on a rainy day are one-in-a-million, if it rains an average of three days per week and if you play golf every Saturday & Sunday — rain or shine — what are your chances of being struck by lightning in a given week? (Assume lightning only strikes on rainy days.) (8) How many ways can Alice, Ben, Charles and Danièle arrange themselves around a square table? How many ways can they arrange themselves around a circular table? (9) How many arrangements of rider & driver are there if Alice, Ben, Charlie and Danièle pair up on Danièle's motorcycle — one person driving and one person riding behind the driver while holding on?  (10) If there are 6 contestants in a contest, in how many potential ways can 1st, 2nd and 3rd prize be awarded? (11) One person per hundred people has infectious foot-in-mouth disease. The probability of a person with infectious foot-in-mouth disease testing positive is 9/10 and the probability of a person who does not have infectious foot-in-mouth disease testing positive is 2/10. What is the probability that a person who tests positive has the disease? (12) If two dice are rolled three times, what is the probability that the two dice will match (i.e., display the same number) on one of the three rolls? (13) There are two hands of cards, one consisting of nine hearts numbered 1 to 9 (ace is 1) and another hand consisting of five diamonds numbered 1 to 5. If there is an equal chance of choosing a card from either hand, what is the probability that an even card will be a heart? (14) If a single die (one dice ) is rolled three times, what is the probability that each roll will result in a number larger than the previous roll? (15) If three dice are thrown (rolled), what are the odds against rolling a total of 10? (16) Out of five men and five women, how many ways are there to form a committee consisting of three women and two men ? (17) How many different poker hands (5 cards) are possible with a standard 52-card deck? (18) If 5 cards are drawn from a deck consisting of only the 13 hearts from a 52-card deck, what is the probability of getting all of the face cards (King, Queen and Jack)? (19) What is the probability of getting a full house in five cards drawn in a poker game from a standard 52-card deck? [A  full house  consists of 3 cards of the same kind (e.g., 3 Queens) and 2 cards of another kind (eg, 2 Aces).] (20) What is the probability of getting a pair in five cards drawn in a poker game from a standard 52-card deck? [A  pair  consists of 2 cards of the same kind (eg, 2 Queens) and 3 cards that are different from the kind of the pair (eg, different from Queens) and that are all different from each other.]   Answers (1) A fruit bowl has 6 apples and 4 oranges. If one piece of fruit is selected at random and then a second piece of fruit is selected at random, what are the chances that both pieces of fruit will be apples? There are six chances in ten that the first choice is an apple. With 5 apples and 4 oranges left in the bowl, there are five chances in nine that the second choice is an apple. The product of the two choices is: 6 5 30 1 --- X --- = --- = --- (one chance in three) 10 9 90 3 (2) Golden Hoof (GH) is twice as likely to win a horse race than Silver Saddle (SS) and Silver Saddle is twice as likely to win as Chomping-at-the-Bit (CB). If only the 3 horses are in the race, what is the probability that Golden Hoof will not win? If the chance that CB will win is 1 then the chance that SS will win is 2 and for GH it is 4. If GH has 4 out of a total of 4 + 2 + 1 = 7 chances of winning, that means 3 out of 7 = 3/7 chances of not winning. (3) Given a hand containing the nine numbered hearts from a deck of cards [2 to 9, with Ace counting as one (1)] — what is the probability of drawing three cards in sequence from the hand such that each selection results in a number larger than the previous one? There are six possible permutations of any three different numbers. Only one of those permutations will be strictly increasing. Therefore, there is only one chance in six (1/6) that any sequential selection of three cards will be strictly increasing. Note that the size of the hand makes no difference to the answer as long as the hand contains three or more cards. Any three distinct numbers have 6 distinct arrangements.  (4) What are the odds against rolling a seven with a single throw of a pair of dice [2 dice]? There are 6 x 6 = 36 possible outcomes, but only the following 6 outcomes produces totals to seven: 1,6 2,5 3,4 6,1 5,2 4,3 That means that there are 36 - 6 = 30 ways in which the two dice can  not  roll a seven. Therefore, the odds against rolling a seven are 30-to-6 or (reducing) 5-to-1 (5) There are six men and twelve women. Half of the men have gray hair and so do half of the women. What are the chances that a person chosen at random will be a man, a person with gray hair or both? One third of the people are men and half have gray hair 1 1 5 --- + --- = --- 3 2 6 But this will double-count men with gray hair (who are in both categories). Three of the eighteen (1/6) are men with gray hair, so 5 1 2 --- - --- = --- 6 6 3 is the chance that a person chosen at random with be either a man or a person with gray hair or both. A simpler solution is to exclude women without grey hair (6/18 excluded, so 12/18 = 2/3 are to be counted.) (6) How many dice must be thrown to give a better than 50% chance that at least one of the dice will show a one (1) (a single snake eye )? If one die is thrown, the chances of not rolling a one (1) are
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