1. **Problem statement:** Aoibhín guesses answers to 20 multiple-choice questions, each with 4 options and only 1 correct. We find probabilities for getting none, exactly five, and four to six correct answers. 2. **Formula:** Use the binomial probability formula: $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $n=20$, $p=\frac{1}{4}=0.25$, and $k$ is the number of correct answers. 3. **(i) Probability of none correct ($k=0$):** $$P(X=0) = \binom{20}{0} (0.25)^0 (0.75)^{20} = 1 \times 1 \times 0.75^{20}$$ Calculate: $$0.75^{20} \approx 0.003$$ 4. **(ii) Probability of exactly five correct ($k=5$):** $$P(X=5) = \binom{20}{5} (0.25)^5 (0.75)^{15}$$ Calculate binomial coefficient: $$\binom{20}{5} = \frac{20!}{5!15!} = 15504$$ Calculate powers: $$0.25^5 = 0.0009765625$$ $$0.75^{15} \approx 0.013363$$ Multiply all: $$P(X=5) = 15504 \times 0.0009765625 \times 0.013363 \approx 0.202$$ 5. **(iii) Probability of 4, 5, or 6 correct:** Calculate each term and sum: - For $k=4$: $$P(X=4) = \binom{20}{4} (0.25)^4 (0.75)^{16}$$ $$\binom{20}{4} = 4845$$ $$0.25^4 = 0.00390625$$ $$0.75^{16} \approx 0.010022$$ $$P(X=4) = 4845 \times 0.00390625 \times 0.010022 \approx 0.190$$ - For $k=5$ (from above): $0.202$ - For $k=6$: $$P(X=6) = \binom{20}{6} (0.25)^6 (0.75)^{14}$$ $$\binom{20}{6} = 38760$$ $$0.25^6 = 0.000244140625$$ $$0.75^{14} \approx 0.017798$$ $$P(X=6) = 38760 \times 0.000244140625 \times 0.017798 \approx 0.168$$ Sum: $$P(4 \leq X \leq 6) = 0.190 + 0.202 + 0.168 = 0.560$$ **Final answers:** (i) $0.003$ (ii) $0.202$ (iii) $0.560$