what work was used or how was technology used to find the provided answer?  The values provided in the Excel spreadsheet are values needed.  However, what other values are needed and how will all be used to find the probability of 6 or more?



To find the probability that six or more customers used their phones for guidance on purchasing decisions, we need to understand the use of the **binomial distribution** and how we calculated the probabilities. Here's a breakdown of the steps and the values required:

### Values Needed:

1. **Probability of Success (p):**  
   This is the probability that a customer uses their phone inside a store for guidance. In this case, \( p = 0.58 \).

2. **Number of Trials (n):**  
   This represents the total number of customers in the sample. Here, \( n = 15 \).

3. **Binomial Coefficients (Combinations):**  
   For each number of successes \( k \) (from 0 to 15), we need the binomial coefficient \( \binom{n}{k} \), which is calculated as:
   \[
   \binom{n}{k} = \frac{n!}{k!(n-k)!}
   \]
   This represents the number of ways \( k \) successes can occur in \( n \) trials.

4. **Probability of Failure (q):**  
   The probability that a customer does **not** use their phone is \( q = 1 - p = 0.42 \).

5. **Individual Binomial Probabilities:**  
   For each number of successes \( k \), the binomial probability \( P(X = k) \) is given by:
   \[
   P(X = k) = \binom{n}{k} p^k q^{n-k}
   \]
   This formula calculates the probability of having exactly \( k \) successes in \( n \) trials.

### How All Values Are Used to Find the Probability of 6 or More:

To find the probability that **six or more** customers used their phones, we do the following:

1. **Calculate Binomial Probabilities for \( k = 0 \) to \( k = 5 \):**  
   We calculate \( P(X = k) \) for each value of \( k \) from 0 to 5 using the binomial probability formula. These probabilities represent the scenarios where fewer than 6 customers use their phones.

2. **Compute the Cumulative Probability for \( X < 6 \):**  
   Sum the probabilities calculated in step 1:
   \[
   P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
   \]

3. **Determine the Complement for \( X \geq 6 \):**  
   Since we want the probability of 6 or more, we use the complement rule:
   \[
   P(X \geq 6) = 1 - P(X < 6)
   \]
   This gives us the probability of having 6 or more successes.

### Using Excel:

If you are using Excel to calculate this, you can use the `BINOM.DIST` function for the probabilities. Here’s how you might approach it:

- **Step 1:** Calculate individual probabilities for \( P(X = k) \) where \( k = 0 \) to \( k = 5 \) using the formula:
  ```excel
  =BINOM.DIST(k, n, p, FALSE)
  ```
  where `k` is the number of successes, `n` is the number of trials, `p` is the probability of success, and `FALSE` indicates you want the probability for an exact number.

- **Step 2:** Sum these probabilities to get \( P(X < 6) \):
  ```excel
  =SUM(BINOM.DIST(0, 15, 0.58, FALSE), BINOM.DIST(1, 15, 0.58, FALSE), ..., BINOM.DIST(5, 15, 0.58, FALSE))
  ```

- **Step 3:** Calculate the probability of 6 or more:
  ```excel
  =1 - [P(X < 6)]
  ```

These steps illustrate the logic behind computing the desired probability using both manual calculations and technology tools like Excel.

Would you like further clarification, or do you have other questions?

Here are some related questions you might explore:

1. How would you calculate the exact probability for \( X = 7 \) using Excel?
2. What is the probability that **exactly** half of the customers use their phones?
3. How can technology tools like Excel help in calculating probabilities for large sample sizes?
4. What is the expected number of customers using phones if the probability changes to 0.65?
5. How would the approach differ if this was a continuous probability distribution?

**Tip:** When using tools like Excel, familiarize yourself with built-in functions like `BINOM.DIST`, as they simplify complex statistical calculations.

Learn how to calculate probabilities using the binomial distribution with a step-by-step guide. Understand the concepts of probability theory, binomial coefficients, and binomial probability formulas applied to real-world scenarios. Suitable for advanced high school and undergraduate students.

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