Help me answer questions 18 to 24

III]. A multiple-choice quiz has 4 questions, each with 4 choices. If a student picks every answer at random, what is the probability that they get all the answers correct? 21. There are seven people in a club: A, B, C, D, E, F, and G. A oommittee of 3 people is formed. What is the probability that G is on the committee? (Hint: what is the total number of committees possible and what is the total number of committees possible with G as a member?) 22. LetA be a set with 9 distinct elements and let 504) be its power set. What is the probability that if one set is chosen from the power set, it will have fewer than 3 elements (that is, D, 1, or 2 elements]? 23.Each die has 6 faces with 1, 2, 3, 4, 5, or E dots. Suppose two dice are rolled. [2 points each} a] Let (a,b) denote the number of dots on the top face [red die, yellow die}. There are 35 possible ordered pairs. How many different sums are possible if sum = a + b? b] The sums do not occur with the same frequency. Which sum is the most likely to oocur? Explain. 24. The Birthday Problem is a famous problem that gives you the probability that in a group of n people at least two have the same birthday. You will need to do some research online. [2 points each} a] Let P = probability that at least two people have the same birthday, and let n be the number of people. What formula gives you P in terms of n? b} What is the probability that in a group of 30 people at least two have the same birthday? cl What is the number of people n for which the probability is at least 99% of at least two people with the same birthday? dj Based on what you have learned, what is the probability that in a group of 5 people, at least two were born in the same month? Show formula you used to determine your answer. Approximate answer to nearest tenth. This assignment is a basic introduction to combinatorics and probability. Before you begin, you will need to do some independent research. Do not attempt the problems without first looking up the following topics and taking some notes for yourself. You can use the textbook (primarily Chapter 9) and you can also do research online. Please let me know if you have any questions as you progress through the project, especially if are not sure how to interpret a problem. Topics: . Meaning of n! (n factorial) . Basic Counting Principal (or Multiplication Rule) . Definition of permutation . Definition of combination . The formula for ner . The formula for ncr, which is also represented as () . Number of arrangements of n distinct objects in a row . Number of arrangements of n distinct objects in a circle . Number of ways we can arrange n objects where r, are identical, 72 are identical, ..., and rx are identical. . Pascal's triangle . Number of subsets in a set with n elements . What is a sample space? . What is the definition of simple probability? Between what two values does probability lie? . What are mutually exclusive events? . What is the probability of event A or event B, if the events are mutually exclusive? What are independent events? . What is the probability of event A and event B, if the events are independent?the in Etween 100 and 99g odd lights? For exan ould be examples of such elements, Explain prepare the sar gents as there are with 5 eler An n-bit string is g of length n, that is it is string of ne ymb of which is a 0 a The weight of a bit string ng is the number of 1's contain How many 8-bit strings have weight equal to a tion for 71: tion 14 Simplify the expression; n!(n-1 (n-2)! 15 coin is tossed times. What the probab ity of obtaining exactly 2 tails? int: make the ample space for tossing a coin three times 16. A true-fals hsis of 5 questions What is the probability answer or all false? (Hint: make a sample space.) lity tick rked If one ticket is multiple of 3? 18. A basket contains 12 red balls and 8 yellow balls. One ball is picked at random and not replaced, and then a second ball is picked at random from the remaining balls. What is the probability of a red ball followed by a yellow ball? 19. Four toy blocks each with one of the letters M, A, T, and H are placed in a box. If you randomly pick them one at a time without replacement, what is the probability that you will choose them in the order MATH