Multiple Choice

1. A fair coin is tossed twice. What is the probability of getting two heads? Use a tree diagram to visualize the outcomes.

   $3/4$

   $1/2$

   $1/3$

   $1/4$

2. A bag contains $3$ red marbles and $2$ blue marbles. A marble is drawn, not replaced, and then a second marble is drawn. What is the probability that both marbles drawn are red?

   $1/4$

   $9/25$

   $3/5$

   $3/10$

3. A student takes a bus to school $60\%$ of the time and walks $40\%$ of the time. If they take the bus, the probability of being late is $0.1$. If they walk, the probability of being late is $0.2$. What is the probability that the student is late for school on any given day?

   $0.15$

   $0.2$

   $0.14$

   $0.1$

4. A box contains $5$ green and $4$ yellow balls. Two balls are drawn without replacement. If the first ball drawn is yellow, what is the probability that the second ball drawn is also yellow?

   $3/8$

   $4/9$

   $1/2$

   $1/3$

5. A biased coin lands on heads with a probability of $0.6$. If the coin is tossed three times, what is the probability of getting exactly two heads?

   $0.216$

   $0.432$

   $0.432$

   $0.36$

6. In a factory, Machine A produces $70\%$ of the items, and Machine B produces $30\%$. Machine A produces $5\%$ defective items, while Machine B produces $10\%$ defective items. If an item is chosen at random and found to be defective, what is the probability that it was produced by Machine A?

   $6/13$

   $7/13$

   $1/2$

   $3/13$

7. A tree diagram shows that the probability of event A occurring is $P(A) = 0.4$. If A occurs, the probability of event B occurring is $P(B|A) = 0.3$. What does the product $P(A) \times P(B|A)$ represent?

   The probability of both A and B occurring.

   The probability of A or B occurring.

   The probability of event B occurring.

   The probability of event A occurring given B.

8. A student has a $0.8$ probability of passing their Math test and a $0.7$ probability of passing their Science test. The outcomes are independent. What is the probability that the student passes at least one of the two tests?

   $0.56$

   $0.94$

   $0.24$

   $0.06$

9. A bag contains only red and blue balls. The probability of drawing a red ball first is $0.6$. After drawing a red ball, the probability of drawing another red ball is $0.5$. If a blue ball is drawn first, the probability of drawing another blue ball is $0.8$. If you draw two balls without replacement, what is the probability of drawing a red ball second, given that the first ball was blue?

   $0.2$

   $0.8$

   $0.6$

   $0.4$

10. In a school, $40\%$ of students study French, and $60\%$ study Spanish. Of those who study French, $75\%$ also study History. Of those who study Spanish, $60\%$ also study History. What is the probability that a randomly selected student studies History?

    $0.7$

    $0.66$

    $0.65$

    $0.45$

11. A basketball player has a $70\%$ chance of making a free throw. If they attempt two free throws, what is the probability that they make exactly one free throw? Assume independence.

    $0.49$

    $0.42$

    $0.7$

    $0.21$

12. A spinner has $4$ equal sections: Red, Blue, Green, Yellow. It is spun three times. What is the probability of spinning Red, then Blue, then Green in that specific order?

    $3/4$

    $1/4$

    $1/64$

    $3/64$

13. When drawing a tree diagram for two events, A and B, how would you know if events A and B are independent?

    The product of probabilities along any path is always $1$.

    The conditional probability of B given A, $P(B|A)$, is different from $P(B)$.

    The conditional probability of B given A, $P(B|A)$, is equal to $P(B)$.

    The number of branches for A is equal to the number of branches for B.

14. A diagnostic test for a disease is $95\%$ accurate (true positive and true negative). $1\%$ of the population has the disease. If a person tests positive, what is the probability they actually have the disease? (Use a tree diagram).

    $0.059$

    $0.161$

    $0.01$

    $0.95$

15. A student takes two quizzes. The probability of passing the first is $0.7$. If they pass the first, the probability of passing the second is $0.8$. If they fail the first, the probability of passing the second is $0.5$. What is the probability that the student fails the second quiz?

    $0.14$

    $0.15$

    $0.29$

    $0.71$

16. A class has $15$ boys and $10$ girls. Two students are chosen randomly without replacement to represent the class. What is the probability that one boy and one girl are chosen?

    $1/2$

    $1/4$

    $1/5$

    $3/10$

17. When constructing a tree diagram for two sequential events, what is the sum of the probabilities of all possible final outcomes (at the end of all branches)?

    It depends on the number of events.

    It is always $1$.

    It is always $0$.

    It is always $0.5$.

18. A tree diagram shows two events, E and F. $P(E) = 0.6$, and $P(F|E) = 0.7$. What is the probability that both E and F occur, i.e., $P(E \text{ and } F)$?

    $0.7$

    $0.24$

    $0.42$

    $0.6$

19. A student has three options for transport: Bus ($50\%$), Car ($30\%$), or Bicycle ($20\%$). The probability of being late is $0.1$ for Bus, $0.05$ for Car, and $0.2$ for Bicycle. What is the probability that the student takes the Car AND is NOT late?

    $0.3 \times 0.95$

    $0.3 \times 0.05 + 0.7 \times 0.3$

    $0.3 \times 0.05$

    $0.3 \times 0.95 + 0.05$

20. A bag contains $5$ red and $5$ blue balls. You draw two balls. Which of the following scenarios would require a tree diagram where the probabilities on the second set of branches CHANGE based on the first draw?

    Drawing two balls without replacement.

    Drawing a red ball, then drawing a blue ball with replacement.

    Drawing two balls with replacement.

    Flipping a coin twice.