1. **Problem Statement:** We have two binomial probability problems involving quizzes where a student guesses answers. 2. **Binomial Probability Formula:** The probability of exactly $k$ successes in $n$ trials is given by: $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $p$ is the probability of success on a single trial. --- ### Problem 9: True/False Quiz (4 questions) 3. **a. Tree Diagram Probabilities:** Each question has 2 outcomes: Correct (C) or Wrong (W). Probability of Correct $p=\frac{1}{2}$, Wrong $1-p=\frac{1}{2}$. The tree has $2^4=16$ outcomes, each with probability $\left(\frac{1}{2}\right)^4=\frac{1}{16}$. 4. **b. Probability of EXACTLY 75% Correct (3 out of 4):** Number of ways to get exactly 3 correct answers: $$\binom{4}{3} = 4$$ Probability: $$P(X=3) = 4 \times \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^1 = 4 \times \frac{1}{8} \times \frac{1}{2} = 4 \times \frac{1}{16} = \frac{4}{16} = \frac{1}{4}$$ 5. **c. Using combinations or calculator:** Same as above, using formula: $$P(X=3) = \binom{4}{3} \left(\frac{1}{2}\right)^3 \left(1-\frac{1}{2}\right)^{1} = \frac{1}{4}$$ --- ### Problem 10: Multiple Choice Quiz (4 questions, 4 choices each) 6. **a. Tree Diagram Probabilities:** Each question has 2 outcomes: Correct (C) or Wrong (W). Probability of Correct $p=\frac{1}{4}$, Wrong $1-p=\frac{3}{4}$. The tree has $2^4=16$ outcomes, each with probability $p^k (1-p)^{4-k}$ depending on number of correct answers $k$. 7. **b. Probability of EXACTLY 50% Correct (2 out of 4):** Number of ways: $$\binom{4}{2} = 6$$ Probability: $$P(X=2) = 6 \times \left(\frac{1}{4}\right)^2 \times \left(\frac{3}{4}\right)^2 = 6 \times \frac{1}{16} \times \frac{9}{16} = 6 \times \frac{9}{256} = \frac{54}{256} = \frac{27}{128} \approx 0.2109$$ 8. **c. Probability of EXACTLY 0% Correct (0 out of 4):** Number of ways: $$\binom{4}{0} = 1$$ Probability: $$P(X=0) = 1 \times \left(\frac{1}{4}\right)^0 \times \left(\frac{3}{4}\right)^4 = 1 \times 1 \times \left(\frac{81}{256}\right) = \frac{81}{256} \approx 0.3164$$