1. **Stating the problem:** We want to calculate the probability of exactly 6 successes in 25 trials with success probability $p=0.26$ using the binomial formula. 2. **Formula used:** The binomial probability formula is $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $n=25$, $k=6$, and $p=0.26$. 3. **Calculate the binomial coefficient:** $$\binom{25}{6} = \frac{25!}{6! \cdot (25-6)!}$$ 4. **Calculate the probability:** $$P(X=6) = \binom{25}{6} \times 0.26^6 \times (1-0.26)^{19}$$ 5. **Intermediate step showing cancellation:** $$P(X=6) = \binom{25}{6} \times 0.26^6 \times 0.74^{19}$$ 6. **Numerical approximation:** $$P(X=6) \approx 0.18 = 18\%$$ 7. **Additional value $z$ given:** $$z \approx 0.053 \approx 0.15\%$$ This completes the calculation of the binomial probability for $k=6$ successes out of $n=25$ trials with $p=0.26$. The table described relates to test results for pollen allergy, showing probabilities of positive and negative test outcomes.