1. **Problem Statement:** Calculate the probability that between 4 or 5 beta testers out of 18 successfully adopt a new software feature, given the success probability $p=0.25$ per tester. 2. **Formula and Explanation:** This is a binomial probability problem where the number of trials $n=18$, success probability $p=0.25$, and failure probability $q=1-p=0.75$. The binomial probability formula is: $$P(X = k) = \binom{n}{k} p^k q^{n-k}$$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. 3. **Calculate $P(X=4)$:** $$P(X=4) = \binom{18}{4} (0.25)^4 (0.75)^{14}$$ Calculate $\binom{18}{4}$: $$\binom{18}{4} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2 \times 1} = 3060$$ So, $$P(X=4) = 3060 \times (0.25)^4 \times (0.75)^{14}$$ 4. **Calculate $P(X=5)$:** $$P(X=5) = \binom{18}{5} (0.25)^5 (0.75)^{13}$$ Calculate $\binom{18}{5}$: $$\binom{18}{5} = \frac{18 \times 17 \times 16 \times 15 \times 14}{5 \times 4 \times 3 \times 2 \times 1} = 8568$$ So, $$P(X=5) = 8568 \times (0.25)^5 \times (0.75)^{13}$$ 5. **Calculate the total probability for 4 or 5 successes:** $$P(4 \text{ or } 5) = P(X=4) + P(X=5)$$ 6. **Numerical evaluation:** Calculate powers: $$(0.25)^4 = 0.00390625, \quad (0.75)^{14} \approx 0.013363$$ $$(0.25)^5 = 0.0009765625, \quad (0.75)^{13} \approx 0.017817$$ Calculate each term: $$P(X=4) = 3060 \times 0.00390625 \times 0.013363 \approx 0.1597$$ $$P(X=5) = 8568 \times 0.0009765625 \times 0.017817 \approx 0.1491$$ 7. **Final answer:** $$P(4 \text{ or } 5) = 0.1597 + 0.1491 = 0.3088$$ **Therefore, the probability that between 4 or 5 beta testers adopt the feature is approximately 0.309.**