Exercise on Counting Techniques and Probability:

  1. How many ways can the five letters in the word “BASIC” be rearranged to form a new five-letter code word?
  1. If one wishes to randomly select a representative from a group 50 individuals which including Paul, what is probability that Paul will be the selected representative for this group of 50 people?
  1. In drawing a set of 5 cards from a deck 52 cards for playing poker game, how many ways can it be “three of a kind”?
  1. There are 3 male students and 4 female students in a class. How many ways can we select two students, one male and one female, to represent the class for a meeting?
  1. There are 3 male students and 4 female students in a class. How many ways can we form a committee by selected 2 male and 3 female students to represent the class for a meeting?
  1. How many ways can a person select 3 paintings from a selections of 10 different paintings?
  1. How many ways can you select 3 pictures from a collection of 10 pictures and hang them on the wall in a row? (Different arrangements are considered as different outcomes.)
  1. How many ways can the eleven letters in the word “MATHEMATICS” be rearranged to form a new eleven-letter code word?
  1. You are planning a trip to visit 3 countries in Europe and you need to choose these 3 countries from a list of 10 countries. How many ways can you choose these 3 countries?
  1. You are planning a trip to visit 3 countries in Europe and you need to choose these 3 countries from a list of 10 countries and decide on the order of the countries to be visited. Different orders of countries to be visited are considered as different travel plans. Each country will be visited once. How many different ways can you plan the trip?
  1. If events A,B, and C are three events in the same space, andA and C are mutually exclusive, P(A) = .4, P(B) = .2, P(C) = .5, P(BC) = .2, find

a)P(B C) =

b)P(AC) =

c)P(B | C) =

d)Are B and C independent, and why?

e)Are A and C independent, and why?

  1. To launch a space shuttle, four booster rockets are needed. If the probability of failure of any of these rockets is .02 and assume that the failures of these booster rockets are mutually independent events. Find the probability that the space shuttle will be launched successfully without any problems from these booster rockets.
  1. An urn contains 6 red balls and 8 white balls.

a)If four balls are to be selected at random with replacement (each time the selected ball is placed back into the urn before selecting another one), what is the probability that there will be 1 red and 3 white balls?

b)If four balls are to be selected at random without replacement (each time the selected ball is not placed back into the urn before selecting another one), what is the probability that there will be 1 red and 3 white balls?

  1. There is a new diagnostic test for testing quality of computer chip. It is designed to be used for a manufacture which has about 0.05% of defective chips. The test is not perfect but will detect a bad computer chip 95% of the time. It will, however, say that a good computer chip is bad about 4% of the time. If a computer chip produced by this manufacture is selected at random and the test indicates that it is bad, what is the probability that this selected computer chip is really a bad chip?