. Imagine you're conducting a survey to determine the probability of a person buying a new iPhone after viewing an advertisement for the updated iPhone. Based on past data, you know that historically 30% of people who view this advertisement end up buying a new iPhone. You're interested in the probability that out of 10 people who view the advertisement, more than 7 will buy a new iPhone

To determine the probability that more than 7 out of 10 people will buy a new iPhone after viewing an advertisement, given that historically 30% of people who see the ad make a purchase, we can model this situation using a **binomial distribution**.

### Problem Setup:
- **n** = 10 (the number of people who view the ad),
- **p** = 0.30 (the probability that a single person buys the iPhone after viewing the ad),
- We need to find the probability that **more than 7** people (i.e., 8, 9, or 10 people) buy the iPhone.

### Binomial Probability Formula:
For a binomial distribution, the probability of getting exactly \( k \) successes (people buying the iPhone) is given by:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n-k}
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient, which is calculated as \( \frac{n!}{k!(n-k)!} \),
- \( p \) is the probability of success (in this case, 0.30),
- \( n \) is the number of trials (10 people),
- \( k \) is the number of successes we are interested in.

### Steps to Solve:
We need to find \( P(X > 7) \), which means:

\[
P(X > 7) = P(X = 8) + P(X = 9) + P(X = 10)
\]

Let’s calculate each term individually. I'll do the computation to sum up these probabilities.The probability that more than 7 out of 10 people will buy a new iPhone after viewing the advertisement is approximately **0.00159**, or about **0.16%**.

Would you like further details or clarifications?

Here are 5 related questions to expand on this:
1. How does the probability change if 15 people view the advertisement instead of 10?
2. What is the probability that exactly 5 people will buy the iPhone?
3. Can you explain the concept of a binomial distribution in more detail?
4. How would the probability change if the historical data showed 50% of viewers bought the iPhone?
5. How does the binomial distribution relate to the normal distribution?

**Tip:** In a binomial distribution, as the number of trials increases, the distribution starts to resemble a normal distribution, especially when \( np \) and \( n(1-p) \) are both large.

Imagine you're conducting a survey to determine the probability of a person buying a new iPhone after viewing an advertisement for the updated iPhone. Based on past data, you know that historically 30% of people who view this advertisement end up buying a new iPhone. You're interested in the probability that out of 10 people who view the advertisement, more than 7 will buy a new iPhone.

Calculate the probability that more than 7 out of 10 people will buy a new iPhone after viewing an advertisement, using the binomial distribution with a 30% success rate. This example explores key probability concepts, such as binomial distribution, the binomial probability formula, and statistical analysis.

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