1. **State the problem:** A test has 10 true/false questions. The student passes by answering at least 8 correctly. We want the probability of passing by guessing. 2. **Formula:** The number of correct answers follows a binomial distribution with parameters $n=10$ and $p=0.5$ (guessing probability). The probability of exactly $k$ correct answers is: $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ 3. **Calculate probability of passing:** Passing means $X \geq 8$, so: $$P(X \geq 8) = P(X=8) + P(X=9) + P(X=10)$$ 4. **Calculate each term:** $$P(X=8) = \binom{10}{8} (0.5)^8 (0.5)^2 = \binom{10}{8} (0.5)^{10}$$ $$P(X=9) = \binom{10}{9} (0.5)^9 (0.5)^1 = \binom{10}{9} (0.5)^{10}$$ $$P(X=10) = \binom{10}{10} (0.5)^{10} (0.5)^0 = \binom{10}{10} (0.5)^{10}$$ 5. **Calculate binomial coefficients:** $$\binom{10}{8} = 45, \quad \binom{10}{9} = 10, \quad \binom{10}{10} = 1$$ 6. **Sum probabilities:** $$P(X \geq 8) = (45 + 10 + 1) (0.5)^{10} = 56 \times \frac{1}{1024} = \frac{56}{1024}$$ 7. **Simplify fraction:** $$\frac{56}{1024} = \frac{7}{128} \approx 0.0547$$ **Final answer:** The probability the student passes by guessing is approximately **0.055**.