Question 1192603
**1. Define the Probability of a Family Member Being Called**

* **Probability of a specific family member being called on a single attempt:** 5/655 

**2. Model the Situation**

* This scenario can be modeled as a binomial distribution.
    * **Binomial Distribution:** Describes the probability of getting *k* successes in *n* independent trials, where each trial has two possible outcomes (success or failure) and the probability of success is constant.

* **In this case:**
    * **Success:** A family member's phone number is called.
    * **Failure:** A family member's phone number is not called.
    * **Number of trials (n):** 519 calls
    * **Probability of success (p):** 5/655

**3. Calculate the Probability of Exactly 3 Calls**

* Use the binomial probability formula:

   P(X = k) = (nCk) * p^k * (1-p)^(n-k)

   where:
       * P(X = k) is the probability of k successes
       * nCk is the number of combinations of n items taken k at a time 
       * p is the probability of success
       * (1-p) is the probability of failure

* For exactly 3 calls:

   P(X = 3) = (519C3) * (5/655)^3 * (650/655)^(519-3)

   * Calculate (519C3) using the combination formula: 
       * (519C3) = 519! / (3! * (519-3)!) 

* **Use a calculator or statistical software to compute the final probability.**

**4. Calculate the Probability of No Less Than 3 Calls**

* This means we need to find the probability of 3 calls, 4 calls, or all 5 family members being called.

* P(X >= 3) = P(X = 3) + P(X = 4) + P(X = 5) 

* Calculate P(X = 4) and P(X = 5) using the binomial probability formula as shown above.

* Sum the probabilities of P(X = 3), P(X = 4), and P(X = 5) to get the probability of no less than 3 calls.

**Important Notes:**

* This calculation assumes that each phone call is independent.
* This model simplifies the real-world scenario, as phone numbers might be called multiple times.

**Disclaimer:** This is a complex calculation. For accurate results, it is recommended to use a statistical software package or a calculator with built-in binomial distribution functions.