Question 1200130
**a) Exactly 26 of them survive their first year of life**

* This is a binomial probability problem.
* **n = 34** (number of trials - eagles)
* **p = 0.72** (probability of success - survival)
* **x = 26** (number of successes)

* **Formula:** P(X = x) = (nCx) * p^x * (1-p)^(n-x) 
    * where nCx = n! / (x! * (n-x)!) 

* **Calculation:** 
    * P(X = 26) = (34C26) * (0.72)^26 * (0.28)^(34-26) 
    * P(X = 26) ≈ 0.116

**b) At most 26 of them survive their first year of life**

* This means we need to find the probability of 26 or fewer eagles surviving.
* P(X ≤ 26) = P(X = 0) + P(X = 1) + ... + P(X = 26)

* This can be calculated using a binomial probability table, a calculator with a binomial probability function, or statistical software. 
* **Using a calculator or software:** P(X ≤ 26) ≈ 0.874

**c) At least 24 of them survive their first year of life**

* This means we need to find the probability of 24 or more eagles surviving.
* P(X ≥ 24) = P(X = 24) + P(X = 25) + P(X = 26) + ... + P(X = 34)

* **Using a calculator or software:** P(X ≥ 24) ≈ 0.878

**d) Between 19 and 26 (including 19 and 26) of them survive their first year of life**

* This means we need to find the probability of 19, 20, 21, ..., 26 eagles surviving.
* P(19 ≤ X ≤ 26) = P(X = 19) + P(X = 20) + ... + P(X = 26)

* **Using a calculator or software:** P(19 ≤ X ≤ 26) ≈ 0.820

**In summary:**

* a) P(X = 26) ≈ 0.116
* b) P(X ≤ 26) ≈ 0.874
* c) P(X ≥ 24) ≈ 0.878
* d) P(19 ≤ X ≤ 26) ≈ 0.820

**Note:** These calculations require the use of a calculator or statistical software with binomial probability functions.