PART 1 For all problems show full working. 1. The menu in a restaurant lists four choices of entrée, seven of main course and three of dessert. Find the number of choices of meal possible: (a) if one of each of the three courses must be chosen (b) if you choose to omit the entrée. 2. In how many ways can 3 different consonants and 4 different vowels be arranged in a row (a) if there are no restrictions (b) if the 4 vowels are always together (c) if the first and last places are occupied by consonants? 3. The digits 3, 5, 7, 8, and 9 are to be used to form numbers. Repetition of a digit is not allowed. (a) How many 5 digit numbers can be formed? (b) How many even four digit numbers can be formed? (c) How many numbers less than 8000 can be formed? 4. The letters forming the word DEFINITION are randomly arranged in a row. (a) How many arrangements can be made if there are no restrictions? (b) How many arrangements are there if the I’s are kept together? 5. If 9 balls numbered 1 to 9 are in a bag and one is selected after the other, without replacement, what is the probability that they are selected in numerical order? 6. The letters of the word POTATO are arranged in a row. What is the probability that the two letters O are together? 7. The life expectancy of a goldfish kept in a fish tank in a person’s home is known to depend on the cleanliness of its water. Pollutants that result from smoking have a huge impact on their life expectancy, as minor particles that land on the surface of the water are extremely toxic to the fish. The probability of a goldfish surviving 4 weeks in a home in which no-one smokes is 0.9, whereas the probability of a goldfish surviving 4 weeks in a home in which smokers are present is 0.3. If there is a 40% chance of selecting a home in which smokers are present, and a goldfish is placed in a tank in a random house, what is the probability of it surviving 4 weeks? (e)Check the previous answer using a tree diagram. (f)Find the probability that Leisa wins the: (i) 4th game(ii)5th game(iii)10th game PART 2 For all problems show full working. 1. A committee of 6 is to be selected from 11 people of whom Alain and Bernhard are two. (a) How many different committees can be formed? (b) How many committees contain both Alain and Bernhard? 2. In how many ways can a committee of 8 be chosen from 5 English teachers, 4 Maths teachers and 3 Economics teachers? (a) if there are no restrictions (b) if each committee contains exactly 2 Maths teachers? (c) if each committee contains at least 3 English teachers and at least one teacher of each of the other subjects? 3. From a bag containing 4 red and 5 blue balls, 3 are selected without replacement. What is the probability all are blue? 4. How many varieties of fruit salad, using at least two fruits, can be obtained from apples, oranges, pears, passionfruit, kiwi fruit and nectarines, taken any number at a time? 5. There are five vowels and 21 consonants in the English alphabet. A young child has a play set containing one of each of the letters. She randomly selects, without replacement, four letters to make a ‘word’. (a) Find the probability distribution for the number of vowels in her ‘word’. (b) What is the probability that her ‘word’ contains at least two vowels? 6. Tattslotto: A player picks a selection of six numbers from the numbers 1 to 45. To determine the number of winners eight numbers are chosen at random – the first six are designated as the winning numbers, and the other two as the supplementary numbers. Prizes are as follow: Division 1: 6 winning numbers Division 2: 5 winning numbers and one supplementary Division 3: 5 winning numbers Division 4: 4 winning numbers Division 5: 3 winning numbers and 1 supplementary Find the probabilities of winning each division. 7. Sue is notorious for being late for work, but she really does try to get there on time. When she is late one day then she makes a real effort not to be late the next day and theprobability is 0.2. If she is on time on one day then she is highly likely to be late the nextday and the probability is 0.9. (a) What is the transition matrix? (b) Sue is late on Monday. (i) What is the initial state matrix? (ii) What is the probability that she is late on Friday? (c) Sue is on time on Monday. (i)What is the initial state matrix? (ii)What is the probability that she is late on Friday? (d) Assume that there is a fifty-fifty chance that she is late on Monday. (i)What is the initial state matrix? (ii)What is the probability that she is late on Friday?