Multiple Choice

1. A fair coin is flipped twice. What is the probability of getting two heads?

   $3/4$

   $1/8$

   $1/2$

   $1/4$

2. Events $A$ and $B$ are independent. If $P(A) = 0.6$ and $P(B) = 0.5$, what is $P(A \text{ and } B)$?

   $0.1$

   $0.3$

   $1.1$

   $0.2$

3. A bag contains $3$ red marbles and $2$ blue marbles. A marble is drawn, its color is noted, and then it is replaced. A second marble is drawn. What is the probability that both marbles drawn are red?

   $3/5$

   $9/25$

   $6/25$

   $3/10$

4. A fair six-sided die is rolled, and then a fair coin is flipped. How many possible outcomes are there in the sample space for this sequence of events?

   $2$

   $12$

   $8$

   $6$

5. Let $P(X \text{ and } Y) = 0.2$ and $P(Y) = 0.5$. What is the conditional probability $P(X|Y)$?

   $0.4$

   $0.7$

   $0.2$

   $0.3$

6. In a class, $40\%$ of students study Math, and $30\%$ of students study Science. If studying Math and studying Science are independent events, what is the probability that a randomly chosen student studies both Math and Science?

   $0.1$

   $0.4$

   $0.12$

   $0.7$

7. A tree diagram starts with two branches: one for ``Rain'' (probability $0.3$) and one for ``No Rain'' (probability $0.7$). From the ``Rain'' branch, there are two sub-branches: ``Traffic Jam'' (probability $0.8$) and ``No Traffic Jam'' (probability $0.2$). What is the probability of it raining AND there being a traffic jam?

   $0.06$

   $0.11$

   $0.24$

   $0.5$

8. A box contains $4$ apples and $6$ oranges. A fruit is chosen at random and eaten. Then a second fruit is chosen at random. What is the probability that the first fruit was an apple and the second fruit was an orange?

   $24/90$

   $24/100$

   $10/19$

   $12/90$

9. Two events $A$ and $B$ are defined such that $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \text{ and } B) = 0.2$. Are events $A$ and $B$ independent?

   Yes, because $P(A) + P(B) = P(A \text{ and } B)$.

   Yes, because $P(A|B) = P(B)$.

   Yes, because $P(A) * P(B) = P(A \text{ and } B)$.

   No, because $P(A) * P(B) \neq P(A \text{ and } B)$.

10. A card is drawn from a standard $52$-card deck. What is the probability that the card is a King, given that it is a face card (King, Queen, or Jack)?

    $1/4$

    $1/13$

    $3/13$

    $1/3$

11. A manufacturer finds that $5\%$ of their products are defective. If two products are chosen randomly and independently, what is the probability that at least one of them is defective?

    $0.1$

    $0.0975$

    $0.0025$

    $0.9025$

12. A bag contains $5$ red balls and $3$ blue balls. A ball is drawn, and without replacement, a second ball is drawn. What is the probability that the second ball drawn is blue, given that the first ball drawn was red?

    $5/7$

    $2/7$

    $3/8$

    $3/7$

13. Using a tree diagram, if the first event has probability $P(A)=0.6$ and $P(\text{not } A)=0.4$, and the second event has $P(B|A)=0.3$ and $P(B|\text{not } A)=0.7$, what is the probability $P(A \text{ and } B)$?

    $0.42$

    $0.28$

    $0.46$

    $0.18$

14. A survey found that $60\%$ of students like pizza, $70\%$ like burgers, and $40\%$ like both. What is the probability that a student likes burgers, given that they like pizza?

    $4/7$

    $0.4$

    $0.9$

    $2/3$

15. A student guesses on two multiple-choice questions. Each question has $4$ options, and only one is correct. What is the probability that the student answers both questions correctly?

    $1/2$

    $1/16$

    $3/4$

    $1/8$

16. A fair coin is flipped, and a standard six-sided die is rolled. What is the probability of getting a `head' on the coin and rolling a number greater than 4 on the die?

    $1/2$

    $1/12$

    $1/3$

    $1/6$

17. A student guesses on three multiple-choice questions. Each question has $4$ options, and only one is correct. If the student guesses randomly for each question, what is the probability that they answer all three questions correctly?

    $1/16$

    $1/12$

    $1/4$

    $1/64$

18. A bag contains $5$ red marbles and $3$ blue marbles. Two marbles are drawn from the bag one after the other without replacement. What is the probability that the first marble drawn is red and the second marble drawn is blue?

    $20/56$

    $15/56$

    $12/56$

    $15/64$

19. In a certain town, $40\%$ of the residents are chocolate lovers. Of the chocolate lovers, $70\%$ prefer dark chocolate. Of those who are not chocolate lovers, $20\%$ prefer dark chocolate. If a randomly selected resident prefers dark chocolate, what is the probability that they are a chocolate lover?

    $0.7$

    $0.12$

    $0.4$

    $0.28$

20. A factory produces items using two machines, Machine A and Machine B. Machine A produces $60\%$ of the items, and Machine B produces $40\%$. $2\%$ of the items produced by Machine A are defective, and $3\%$ of the items produced by Machine B are defective. An item is randomly selected from the factory's total output. What is the probability that the item is not defective?

    $0.6$

    $0.976$

    $0.024$

    $0.975$