1. **Problem Statement:** Calculate the probability that between 4 or 5 beta testers out of 18 successfully adopt a new software feature within the first week, given the success probability $p=0.25$. 2. **Formula Used:** This is a binomial probability problem. The probability of exactly $k$ successes in $n$ trials is given by the binomial probability formula: $$P(X=k) = \binom{n}{k} p^k q^{n-k}$$ where $q = 1-p$. 3. **Calculate individual probabilities:** - For $k=4$: $$P(X=4) = \binom{18}{4} (0.25)^4 (0.75)^{14}$$ - For $k=5$: $$P(X=5) = \binom{18}{5} (0.25)^5 (0.75)^{13}$$ 4. **Calculate binomial coefficients:** $$\binom{18}{4} = \frac{18!}{4! \times 14!} = 3060$$ $$\binom{18}{5} = \frac{18!}{5! \times 13!} = 8568$$ 5. **Calculate probabilities:** $$P(X=4) = 3060 \times (0.25)^4 \times (0.75)^{14}$$ $$P(X=5) = 8568 \times (0.25)^5 \times (0.75)^{13}$$ 6. **Evaluate powers:** $$ (0.25)^4 = 0.00390625, \quad (0.25)^5 = 0.0009765625$$ $$ (0.75)^{14} \approx 0.013363, \quad (0.75)^{13} \approx 0.017817$$ 7. **Calculate each term:** $$P(X=4) = 3060 \times 0.00390625 \times 0.013363 \approx 0.1597$$ $$P(X=5) = 8568 \times 0.0009765625 \times 0.017817 \approx 0.1491$$ 8. **Sum probabilities for 4 or 5 successes:** $$P(4 \text{ or } 5) = P(X=4) + P(X=5) = 0.1597 + 0.1491 = 0.3088$$ **Final answer:** The probability that between 4 or 5 beta testers successfully adopt the feature is approximately **0.309**.