Unit 4 - Practice Quiz

1 Which of the following defines the probability of an event in a finite sample space where all outcomes are equally likely?

2 What is the probability of getting a 'Heads' when flipping a single, fair coin?

3 What is the probability of rolling a $4$ on a standard fair six-sided die?

4 If is the probability of an outcome in a sample space , what must be the sum of the probabilities of all outcomes in ?

5 Which of the following represents the valid range for the probability of any event ?

6 If is the probability of an event , what is the probability of its complement, ?

7 If two events and are mutually exclusive (disjoint), what is the probability of their union, ?

8 Which formula correctly represents the Principle of Inclusion-Exclusion for the probability of the union of any two events and ?

9 What is the correct formula for the conditional probability of event given event , assuming ?

10 How is the expression read in the context of probability?

11 If and , what is the value of ?

12 Two events and are defined as independent if and only if which of the following is true?

13 If and are independent events with , what does equal?

14 Events are considered pairwise independent if for all pairs :

15 Does pairwise independence among a set of three events automatically guarantee that they are mutually independent?

16 Which of the following describes a Bernoulli trial?

17 In a Bernoulli trial, if the probability of success is , what is the probability of failure, denoted as ?

18 What is the formula for the binomial distribution probability of getting exactly successes in independent Bernoulli trials?

19 In discrete mathematics, what is a random variable formally defined as?

20 Which of the following is an example of a discrete random variable?

21 A bag contains 4 red balls, 5 blue balls, and 6 green balls. If two balls are drawn at random without replacement, what is the probability that both are blue?

22 A biased die is rolled. The probability of rolling a 6 is three times as likely as rolling any other specific number. What is the probability of rolling an even number?

23 Let and be events such that , , and . What is the probability of the complement of ?

24 A family has two children. Given that at least one of the children is a boy, what is the probability that both children are boys? (Assume boys and girls are equally likely).

25 A fair coin is tossed three times. Let be the event that the first toss is heads, and be the event that there is exactly one head in the three tosses. Are and independent?

26 Suppose a fair 4-sided die with faces numbered 1 to 4 is rolled. Let be the event that the roll is 1 or 2, be the event that the roll is 1 or 3, and be the event that the roll is 1 or 4. Which of the following is true regarding events A, B, and C?

27 A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

28 Let be the random variable representing the number of heads obtained when two fair coins are tossed. What is the expected value of ?

29 From a standard deck of 52 cards, 3 cards are drawn at random without replacement. What is the probability that all 3 cards are hearts?

30 In a class of 100 students, 60 study Mathematics, 50 study Computer Science, and 20 study neither. What is the probability that a randomly chosen student studies both subjects?

31 In a factory, Machine A produces 40% of the items and Machine B produces 60%. Machine A has a 5% defect rate, and Machine B has a 2% defect rate. If a randomly selected item is defective, what is the probability it was produced by Machine A?

32 Events and are independent. If and , what is ?

33 A coin is loaded so that heads is twice as likely to appear as tails. What is the probability of getting tails?

34 A marksman hits a target with a probability of 0.8 on any given shot. What is the probability that he hits the target exactly 4 times out of 5 shots?

35 A random variable has the probability distribution , , and . What is the variance of ?

36 An integer is chosen at random from 1 to 50 inclusive. What is the probability that the number is divisible by 4 or 5?

37 Two fair 6-sided dice are rolled. What is the probability that the sum is 8, given that at least one of the dice shows a 3?

38 For three events , , and to be mutually independent, they must be pairwise independent and satisfy which additional condition?

39 If a student randomly guesses on a 10-question true/false exam, what is the probability of getting at least 9 questions correct?

40 Let be the random variable denoting the sum of the numbers rolled with two fair six-sided dice. What is the value of ?

41 Five individuals attend a party and leave their hats at the coat check. At the end of the party, the hats are returned completely at random. What is the probability that exactly two individuals receive their own hats?

42 Suppose we distribute 5 distinguishable balls into 3 distinguishable boxes uniformly at random. What is the probability that no box remains empty?

43 A random monotonic path on a grid is taken from to , taking steps only right or up . What is the probability that the path never crosses above the diagonal ?

44 A standard six-sided die is loaded such that the probability of rolling a given face is directly proportional to the square of its face value. What is the probability of rolling a prime number?

45 An infinite sample space is assigned probabilities such that for . What is the probability of an outcome being an even number?

46 Given two events and in a sample space, it is known that and . Let be the minimum possible value of and be the maximum possible value of . What is the value of if achieves its absolute minimum?

47 For three mutually independent events , , and , , , and . What is the probability that exactly one of these events occurs?

48 Two cards are drawn sequentially without replacement from a standard 52-card deck. Given that at least one of the cards drawn is an Ace, what is the probability that both cards drawn are Aces?

49 A rare disease affects of the population. A test for the disease has a sensitivity of (true positive rate) and a specificity of (true negative rate). If a randomly selected person tests positive, what is the probability they actually have the disease? (Round to the nearest hundredth of a percent)

50 An urn contains white balls and black balls. A ball is drawn at random and replaced along with additional balls of the same color. A second ball is then drawn. What is the probability that the second ball drawn is white, given that the first ball drawn was white?

51 Events and are independent. It is known that and . What is ?

52 Consider a family with two children. Assume the probability of a child being a boy or girl is equally likely, and genders of children are independent. Let be the event 'the family has children of both sexes' and be the event 'there is at most one girl'. Are and independent?

53 A fair four-sided die with faces numbered 1, 2, 3, and 4 is rolled twice. Let be the event that the first roll is even, be the event that the second roll is even, and be the event that the sum of the two rolls is 5. Which of the following statements is true?

54 Let be an equiprobable sample space. Define , , and . What kind of independence do events , and exhibit?

55 If events are mutually independent, which of the following is absolutely required but is NOT guaranteed by pairwise independence alone?

56 Let be a random variable representing the number of successes in independent Bernoulli trials, each with success probability . What is the probability that is an even number?

57 Consider a redundant system of components where each component fails independently with probability . System A consists of 3 components and works if at least 2 function. System B consists of 5 components and works if at least 3 function. For what values of is System B strictly more reliable than System A?

58 A student guesses on a multiple-choice exam with 100 questions, each having 3 options (only one correct). The number of correct answers follows a binomial distribution . What is the most likely number of correct answers (the mode)?

59 A discrete random variable has the probability mass function for . What is the probability that ?

60 Let and be independent geometric random variables, both representing the number of trials until the first success, with probability of success on each trial. What is ?