SOLUTION: There's a poll going on in a certain town. Out of a list of all phone numbers in town (total 546) the polling organisation chooses 257 random ones. Every phone number is called one
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-> SOLUTION: There's a poll going on in a certain town. Out of a list of all phone numbers in town (total 546) the polling organisation chooses 257 random ones. Every phone number is called one
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Question 1192670: There's a poll going on in a certain town. Out of a list of all phone numbers in town (total 546) the polling organisation chooses 257 random ones. Every phone number is called one by one randomly and same numbers might get called a few times. A family has 4 people with their individual phone numbers. What is the probability of the family receiving exactly 3 phone calls total? And what is the probability of the family receiving no less than 3 calls? Found 2 solutions by CPhill, ikleyn:Answer by CPhill(1987) (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:** 1/546
**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):** 257 calls
* **Probability of success (p):** 1/546
**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) = (257C3) * (1/546)^3 * (545/546)^(257-3)
* Calculate (257C3) using the combination formula:
* (257C3) = 257! / (3! * (257-3)!)
* **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 4 family members being called.
* P(X >= 3) = P(X = 3) + P(X = 4) + P(X = 4)
* Calculate P(X = 4) using the binomial probability formula as shown above.
* Sum the probabilities of P(X = 3) and P(X = 4) 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.
