1. The quiz has 10 questions, each with 6 possible answers, and answers are guessed randomly. 2. Passing requires at least 70% correct, so minimum correct answers to pass is $\lceil 0.7 \times 10 \rceil = 7$. 3. The probability of guessing one question correctly is $\frac{1}{6}$ and incorrectly is $\frac{5}{6}$. 4. The probability of exactly $k$ correct answers out of 10 follows the binomial distribution: $$P(X=k) = \binom{10}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{10-k}$$ 5. The probability of passing is the sum of probabilities of getting 7, 8, 9, or 10 correct answers: $$P(pass) = \sum_{k=7}^{10} \binom{10}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{10-k}$$ 6. Calculate terms: $$P(pass) = \binom{10}{7} \left(\frac{1}{6}\right)^7 \left(\frac{5}{6}\right)^3 + \binom{10}{8} \left(\frac{1}{6}\right)^8 \left(\frac{5}{6}\right)^2 + \binom{10}{9} \left(\frac{1}{6}\right)^9 \left(\frac{5}{6}\right)^1 + \binom{10}{10} \left(\frac{1}{6}\right)^{10}$$ 7. Using the binomial coefficients: $\binom{10}{7} = 120$, $\binom{10}{8} = 45$, $\binom{10}{9} = 10$, $\binom{10}{10} = 1$ 8. Substitute: $$P(pass) = 120 \left(\frac{1}{6}\right)^7 \left(\frac{5}{6}\right)^3 + 45 \left(\frac{1}{6}\right)^8 \left(\frac{5}{6}\right)^2 + 10 \left(\frac{1}{6}\right)^9 \left(\frac{5}{6}\right) + \left(\frac{1}{6}\right)^{10}$$ 9. This value is very small since guessing correctly 7 or more times out of 10 with 1/6 chance is very unlikely. 10. Final answer: $$P(pass) = \sum_{k=7}^{10} \binom{10}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{10-k}$$