Unit 5 - Practice Quiz

1 If an operation can be performed in different ways, and another independent operation can be performed in different ways, what is the total number of ways to perform both operations together?

2 If an event can occur in ways and a second event can occur in ways, but both cannot occur simultaneously, in how many ways can either of the events occur?

3 What is the formula for the number of permutations of distinct objects taken at a time, denoted by ?

4 How many 3-digit numbers can be formed using the digits 1, 2, and 3 without repeating any digit?

5 In how many different ways can the letters of the word 'CAT' be arranged?

6 What is the total number of ways to arrange distinct letters in a row?

7 What is the formula for the number of ways to arrange distinct objects around a circular table?

8 In how many ways can 4 people sit around a circular table?

9 In how many ways can 3 identical letters be dropped into 4 distinct letterboxes?

10 What is the formula for combinations, , representing the number of ways to select objects from distinct objects?

11 In how many ways can a committee of 2 members be selected from a group of 4 people?

12 How many distinct straight lines can be drawn by joining any two points from a set of non-collinear points?

13 Which of the following describes the correct range for the probability of any event ?

14 What is the sum of the probabilities of all the mutually exclusive and exhaustive elementary events of an experiment?

15 If two events and cannot occur at the same time, what are they called?

16 What is the formula for the conditional probability of event given that event has already occurred, denoted as ?

17 If events and are independent, what is equal to?

18 What is the probability of getting a 'Head' when a fair, unbiased coin is tossed once?

19 What is the probability of rolling an even number on a standard 6-sided die?

20 If a single card is drawn from a standard deck of 52 playing cards, what is the probability that it is an Ace?

21 There are 4 different routes to travel from City A to City B, and 3 different routes to travel from City B to City C. In how many ways can a person travel from City A to City C and return back to City A without using the same route twice?

22 How many 4-digit even numbers can be formed using the digits 1, 2, 3, 4, and 5 without repetition?

23 What is the sum of all 3-digit numbers that can be formed using the digits 1, 2, and 3 without repetition?

24 In how many ways can the letters of the word 'SUCCESS' be arranged so that all the 'S's are together?

25 What is the dictionary rank of the word 'LATE' if all permutations of its letters are arranged alphabetically?

26 In how many ways can 4 men and 4 women sit around a circular table such that no two men sit together?

27 5 friends are to be seated around a circular table. In how many ways can they be seated if two specific friends, A and B, refuse to sit next to each other?

28 In how many ways can 8 identical chocolates be distributed among 3 children such that each child receives at least one chocolate?

29 A committee of 4 members is to be formed from 5 men and 4 women. In how many ways can this be done if the committee must contain exactly 2 women?

30 A group of 5 people is to be chosen from 6 men and 4 women. In how many ways can the group be formed so that men are in the majority?

31 How many triangles can be formed by joining 10 points in a plane, if exactly 4 of these points are collinear?

32 Find the number of diagonals in a regular decagon (a polygon with 10 sides).

33 What is the probability that a randomly chosen leap year has 53 Sundays?

34 If A and B are two independent events such that and , what is the value of ?

35 A family has 2 children. Given that at least one of them is a boy, what is the probability that both children are boys?

36 Two standard dice are rolled. Given that the sum of the numbers appearing on them is 8, what is the probability that at least one die shows a 3?

37 Three unbiased coins are tossed simultaneously. What is the probability of getting at most two heads?

38 Two fair dice are thrown together. What is the probability that the product of the numbers obtained is an even number?

39 Two cards are drawn at random from a standard deck of 52 cards without replacement. What is the probability that both cards are Kings?

40 A card is drawn from a well-shuffled pack of 52 cards. What is the probability that it is either a Heart or a Face card?

41 What is the sum of all distinct 5-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition?

42 How many 5-digit numbers divisible by 3 can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition?

43 What is the rank of the word 'TOUGH' if all permutations of its letters are arranged in alphabetical (dictionary) order?

44 In how many ways can the letters of the word 'MISSISSIPPI' be arranged such that no two vowels are adjacent?

45 If 8 distinct people sit around a circular table, in how many ways can they be seated such that two specific people, A and B, do NOT sit diametrically opposite each other?

46 6 men and 6 women sit around a circular table. What is the probability that no two women sit together?

47 In how many ways can 10 identical apples be distributed among 4 children such that each child gets at least 1 apple, but no child gets more than 4 apples?

48 What is the number of arrangements of 6 distinct letters into their respective 6 distinct addressed envelopes such that exactly 2 letters are placed in their correct envelopes?

49 A committee of 6 members is to be formed from 5 teachers and 8 students. Teacher A refuses to be on the committee if Student B is included. In how many ways can the committee be formed?

50 A committee of 5 is formed from 6 men and 4 women. Given that the committee contains at least 1 woman, what is the probability that it contains at least 2 women?

51 What is the maximum number of intersection points produced by 8 circles and 6 straight lines in a plane?

52 How many triangles can be formed by joining the vertices of a regular 12-sided polygon such that no side of the triangle is a side of the polygon?

53 Two real numbers and are chosen uniformly and independently at random from the interval . What is the probability that and ?

54 Let and be two independent events such that and . Find the value of .

55 A diagnostic test for a disease has a 99% accuracy rate for both positive and negative results. The prevalence of the disease in the population is 1 in 1000. If a randomly selected person tests positive, what is the approximate probability they actually have the disease?

56 Urn A contains 3 red and 4 black balls. Urn B contains 5 red and 6 black balls. One ball is transferred from Urn A to Urn B, and then two balls are drawn simultaneously from Urn B. If both drawn balls are red, what is the probability that the transferred ball was red?

57 Three machines A, B, and C produce 40%, 30%, and 30% of a factory's total output, respectively. The defect rates are 2%, 3%, and 4%. If an item is drawn at random and found to be non-defective, what is the probability it was produced by Machine B?

58 A fair coin is tossed 10 times. What is the probability of getting exactly 4 heads, given that at least 2 heads occurred?

59 Three standard 6-sided dice are rolled simultaneously. What is the probability that the sum of the numbers showing on the upper faces is exactly 14?

60 Four cards are drawn at random from a standard 52-card deck. What is the probability of getting exactly two pairs (e.g., two 8s and two Kings)?