Multiple Choice

1. Given two events, A and B, where $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$. Find $P(A \cup B)$.

   0.8

   0.9

   0.7

   0.6

2. If $P(X) = \frac{1}{3}$, $P(Y) = \frac{1}{2}$, and $P(X \cap Y) = \frac{1}{6}$. What is $P(X \cup Y)$?

   $1/3$

   $5/6$

   $2/3$

   $1/2$

3. Suppose $P(M) = 0.6$, $P(N) = 0.3$, and $P(M \cup N) = 0.7$. What is $P(M \cap N)$?

   0.3

   0.1

   0.4

   0.2

4. Two events, A and B, are mutually exclusive. If $P(A) = 0.35$ and $P(B) = 0.2$. What is $P(A \cup B)$?

   0.7

   0.15

   0.55

   0.65

5. In a class, $40\%$ of students like apples, $30\%$ like bananas, and $10\%$ like both. What percentage of students like apples or bananas?

   $60\%$

   $70\%$

   $50\%$

   $80\%$

6. A single card is drawn from a standard deck of $52$ cards. What is the probability of drawing a King or a Spade?

   $1/13$

   $17/52$

   $4/13$

   $16/52$

7. Given $P(E) = \frac{2}{5}$, $P(F) = \frac{1}{2}$, and $P(E \cup F) = \frac{7}{10}$. Find $P(E \cap F)$.

   $1/2$

   $1/10$

   $1/5$

   $3/10$

8. A bag contains $5$ red marbles, $3$ blue marbles, and $2$ green marbles. If one marble is drawn at random, what is the probability that it is red or green?

   $3/10$

   $4/5$

   $1/2$

   $7/10$

9. In a group of $100$ people, $60$ speak English, $30$ speak Spanish, and $10$ speak both. What is the probability that a randomly chosen person speaks English or Spanish?

   0.7

   0.9

   0.6

   0.8

10. When rolling a standard six-sided die, what is the probability of rolling an even number or a number greater than $4$?

    $1/2$

    $2/3$

    $1/6$

    $5/6$

11. Event A has a probability of $0.7$, and event B has a probability of $0.3$. The probability of both A and B occurring is $0.15$. What is the probability of A or B occurring?

    0.85

    0.9

    0.8

    0.75

12. If $P(C) = 0.55$, $P(D) = 0.45$, and $P(C \cup D) = 0.8$. What is $P(C \cap D)$?

    0.3

    0.1

    0.2

    0.15

13. A student is chosen from a class. The probability that the student plays soccer is $0.4$, and the probability that the student plays basketball is $0.3$. If a student cannot play both sports simultaneously, what is the probability that the student plays soccer or basketball?

    0.6

    0.7

    0.8

    0.1

14. In a neighborhood, $50\%$ of households own a dog, $30\%$ own a cat, and $15\%$ own both a dog and a cat. What is the probability that a randomly selected household owns a dog or a cat?

    0.8

    0.65

    0.7

    0.95

15. A card is drawn from a standard $52$-card deck. What is the probability of drawing a red card or a face card (Jack, Queen, King)?

    $32/52$

    $38/52$

    $26/52$

    $8/13$

16. The probability of a student passing Math is $0.7$, and passing Science is $0.6$. The probability of passing both is $0.5$. What is the probability of a student passing Math or Science?

    0.6

    0.7

    0.9

    0.8

17. Given two events A and B. $P(A) = 0.25$, $P(B) = 0.6$, and $P(A \cap B) = 0.1$. Calculate $P(A \cup B)$.

    0.75

    0.85

    0.65

    0.95

18. In a batch of items, the probability of an item having a scratch is $0.15$, the probability of it having a dent is $0.1$, and the probability of it having both a scratch and a dent is $0.05$. What is the probability that an item has a scratch or a dent?

    0.3

    0.25

    0.1

    0.2

19. In a factory, $5\%$ of products are defective due to a manufacturing error (E1), and $3\%$ are defective due to a packaging error (E2). These two types of errors cannot occur on the same product. What is the probability that a product has a manufacturing error or a packaging error?

    0.05

    0.0015

    0.08

    0.03

20. When rolling a six-sided die, what is the probability of rolling a $3$ or an even number?

    $2/3$

    $3/6$

    $1/6$

    $5/6$

21. If $P(A) = 0.5$, $P(B) = 0.4$, and $P(A \cap B) = 0.2$, what is $P(A|B)$?

    $0.4$

    $0.5$

    $0.8$

    $0.2$

22. In a class of 100 students, 60 study Math, 40 study Science, and 20 study both. If a randomly chosen student studies Science, what is the probability that they also study Math?

    $0.2$

    $0.6$

    $0.4$

    $0.5$

23. A standard deck of 52 playing cards is used. If two cards are drawn without replacement, what is the probability that the second card drawn is a King, given that the first card drawn was a King?

    $3/52$

    $1/13$

    $3/51$

    $4/52$

24. In a certain town, $30\%$ of people have red hair and $10\%$ of people have red hair and wear glasses. What is the probability that a randomly chosen person wears glasses, given that they have red hair?

    $0.1$

    $0.3$

    $1/3$

    $0.6$

25. A bag contains 5 red balls and 3 blue balls. If two balls are drawn without replacement, what is the probability that the second ball drawn is blue, given that the first ball drawn was red?

    $3/8$

    $5/7$

    $5/8$

    $3/7$

26. If $P(A) = 0.6$, $P(B) = 0.3$, and $P(B|A) = 0.2$, what is $P(A \cap B)$?

    $0.18$

    $0.3$

    $0.5$

    $0.12$

27. In a class of 25 students, 15 play soccer, 10 play basketball, and 5 play both. If a student is chosen at random and they play soccer, what is the probability that they also play basketball?

    $1/2$

    $1/5$

    $2/5$

    $1/3$

28. Given $P(A) = 0.7$, $P(B) = 0.4$, and $P(A \cap B) = 0.28$. What is $P(A|B)$?

    $0.28$

    $0.4$

    $0.75$

    $0.7$

29. A fair six-sided die is rolled. What is the probability that the result is an even number, given that the result is greater than 3?

    $1/2$

    $1/3$

    $2/3$

    $1/6$

30. In a group of 200 people, 120 like apples, 80 like bananas, and 40 like both. If a person is selected randomly from those who like bananas, what is the probability that they also like apples?

    $40/80$

    $120/200$

    $80/200$

    $40/120$

31. Given two events $A$ and $B$, $P(A) = 0.6$, $P(B) = 0.5$, and $P(A \cup B) = 0.8$. What is $P(A|B)$?

    $0.6$

    $0.8$

    $0.5$

    $0.4$

32. In a school of 100 students, 60 study Math, 40 study Physics, and 20 study both. If a randomly selected student studies Physics, what is the probability they also study Math?

    $\frac{1}{2}$

    $\frac{1}{3}$

    $\frac{1}{4}$

    $\frac{2}{3}$

33. The probability of a student passing Test A is $0.7$, passing Test B is $0.6$, and passing both is $0.4$. If a student passed at least one of the tests, what is the probability they passed Test A?

    $\frac{2}{3}$

    $\frac{4}{9}$

    $\frac{7}{10}$

    $\frac{7}{9}$

34. A standard deck of 52 cards is used. Two cards are drawn randomly without replacement. What is the probability that the second card drawn is a King, given that the first card drawn was a face card (Jack, Queen, or King)?

    $\frac{1}{13}$

    $\frac{11}{153}$

    $\frac{3}{51}$

    $\frac{4}{51}$

35. Two fair six-sided dice are rolled. What is the probability that the sum of the numbers rolled is 7, given that at least one of the dice shows a 3?

    $\frac{1}{6}$

    $\frac{1}{11}$

    $\frac{1}{3}$

    $\frac{2}{11}$

36. Let $A$ and $B$ be two events such that $P(A) = 0.4$, $P(B) = 0.7$, and $P(A \cap B) = 0.3$. What is $P(A'|B)$, where $A'$ is the complement of event $A$?

    $\frac{4}{5}$

    $\frac{2}{5}$

    $\frac{3}{7}$

    $\frac{4}{7}$

37. Events $X$ and $Y$ are independent. $P(X) = 0.6$ and $P(Y) = 0.5$. What is $P(X | Y')$, where $Y'$ is the complement of event $Y$?

    $0.7$

    $0.5$

    $0.6$

    $0.3$

38. In a town, 70% of the people like coffee, 50% like tea, and 30% like both. If a randomly chosen person likes tea, what is the probability that they do not like coffee?

    $0.7$

    $0.3$

    $0.6$

    $0.4$

39. An urn contains 5 red balls and 3 blue balls. Two balls are drawn randomly without replacement. What is the probability that both balls drawn are red, given that at least one of the drawn balls is red?

    $\frac{3}{7}$

    $\frac{1}{2}$

    $\frac{2}{5}$

    $\frac{5}{14}$

40. In a certain population, 2% of people have a rare disease. A test for the disease has a 90% accuracy, meaning if a person has the disease, it tests positive 90% of the time, and if a person does not have the disease, it tests negative 90% of the time. If a randomly selected person tests positive, what is the probability they actually have the disease?

    $\frac{2}{100}$

    $\frac{18}{1000}$

    $\frac{9}{10}$

    $\frac{9}{58}$