What should I buy? A study conducted by the Usie Research Center reported that 58% of customers used their phones inside a store for guidance on purchasing decisions. A sample of 15 customers is studied.
(a). What is the probability that six or more of then used their phones for guidance on purchasing decisions?

To solve this problem, we'll use the **binomial probability formula**, as we're dealing with a fixed number of trials (15 customers) and a probability of success (using their phones) on each trial.

### Step-by-step Solution:

1. **Identify the given values:**
   - Probability of success (using phone) \( p = 0.58 \)
   - Probability of failure \( q = 1 - p = 1 - 0.58 = 0.42 \)
   - Number of trials \( n = 15 \)
   - We want the probability of having **six or more** successes (i.e., \( X \geq 6 \))

2. **Understanding the Binomial Distribution:**
   The binomial distribution gives the probability of having exactly \( k \) successes in \( n \) trials, which is calculated by:

   \[
   P(X = k) = \binom{n}{k} p^k q^{n-k}
   \]

   where \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.

3. **Calculate the cumulative probability:**
   We want to find the probability that six or more customers used their phones. This is:

   \[
   P(X \geq 6) = 1 - P(X < 6)
   \]

   where \( P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) \).

4. **Calculate each of these probabilities using the binomial formula:**

Let's calculate these probabilities using Python.The probability that six or more of the 15 customers used their phones for guidance on purchasing decisions is approximately **0.9521** (or 95.21%).

Would you like more details on the calculation, or do you have any other questions?

Here are some related questions you might consider:

1. What is the probability that exactly 10 customers used their phones for guidance?
2. How would the probability change if the sample size was increased to 20 customers?
3. What is the expected number of customers who use their phones in a sample of 15?
4. How does changing the probability of success \(p\) affect the outcome?
5. What is the variance and standard deviation of the distribution?

**Tip:** Understanding the binomial distribution is useful for analyzing scenarios with fixed trials and two possible outcomes, such as customer behavior studies.

Calculate the probability that six or more out of 15 customers used their phones for guidance on purchasing decisions using the binomial distribution. Understand the binomial probability formula and its application in probability theory.

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