To find the probability that Teesha will not be picked as a guide, we can first determine the total number of ways to pick 3 students from the 26 students in the club.
The total number of ways to choose 3 students from 26 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial.
In this case, \( n = 26 \) and \( r = 3 \):
\[ \binom{26}{3} = \frac{26!}{3!(26-3)!} = \frac{26!}{3! \cdot 23!} \]
Calculating the factorials:
\[ \binom{26}{3} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = \frac{15600}{6} = 2600 \]
So, there are 2600 different ways to choose any 3 students from 26.
Next, we calculate the number of ways to choose 3 students that do not include Teesha. If Teesha is not included, we are left with 25 students. The number of ways to choose 3 students from these 25 is:
\[ \binom{25}{3} = \frac{25!}{3!(25-3)!} = \frac{25!}{3! \cdot 22!} \]
Calculating this gives:
\[ \binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = \frac{13800}{6} = 2300 \]
Now, we can find the probability that Teesha is not picked by dividing the number of ways to choose 3 students from the 25 left by the total number of ways to choose 3 students from the 26:
\[ P(\text{Teesha not picked}) = \frac{\text{Number of ways to choose 3 from 25}}{\text{Number of ways to choose 3 from 26}} = \frac{2300}{2600} = \frac{23}{26} \]
Therefore, the probability that Teesha will not be picked as a guide is
\[ \boxed{\frac{23}{26}}. \]