Question 1192603: 519 numbers are selected randomly from the phone number list, which contains 655 numbers in total. Numbers are dialed one by one and the same number can be redialed. The number of family members is 5 and they all have cell phones. What is the probability that members of this family will be called exactly three times? What is the probability that members of this family will be called at least three times?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **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.
Answer by ikleyn(52803)  (Show Source):
You can put this solution on YOUR website! .
Looking into this problem in the post, I think (it seems to me)
that this composition is totally and absolutely non-sensical as a Math problem.
In other terms, it is a soup of words with no sense.
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