​% of consumers believe that cash will be obsolete in the next 20 years. Assume that 7 consumers are randomly selected. Find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years

To solve this problem, we can use the **binomial probability distribution**. In a binomial distribution, the probability of getting exactly \( k \) successes in \( n \) independent trials is given by the formula:

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

Where:
- \( n \) = number of trials (in this case, 7 consumers)
- \( k \) = number of successes (the number of consumers who believe cash will be obsolete)
- \( p \) = probability of success on a single trial (the proportion of consumers who believe cash will be obsolete)
- \( \binom{n}{k} \) is the binomial coefficient, which is calculated as \( \frac{n!}{k!(n-k)!} \).

However, to calculate the probability that **fewer than 3 consumers** believe that cash will be obsolete, we need to sum the probabilities for 0, 1, and 2 consumers believing cash will be obsolete. So, we need to calculate:

\[
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
\]

### Steps to solve:
1. Determine the value of \( p \) (percentage of consumers who believe cash will be obsolete). This value wasn't provided in the problem, so you might need to provide it.
2. Calculate \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \) using the binomial formula.
3. Add those probabilities to get the final answer for \( P(X < 3) \).

Could you confirm the value of \( p \), or if you need help assuming one based on real-world estimates?

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### Related Questions:
1. How do we find the binomial coefficient \( \binom{n}{k} \) for any values of \( n \) and \( k \)?
2. How do we calculate the cumulative binomial probability for a given \( p \)?
3. What is the relationship between the binomial distribution and normal distribution for large \( n \)?
4. How can we find \( p \) using survey data?
5. What happens to the probability if \( p \) changes significantly?

**Tip:** Always check whether the problem is asking for "fewer than" or "at least" a certain number, as they require different calculations.

% of consumers believe that cash will be obsolete in the next 20 years. Assume that 7 consumers are randomly selected. Find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.

Find the probability that fewer than 3 out of 7 randomly selected consumers believe that cash will be obsolete in the next 20 years using binomial distribution. This problem involves key probability concepts and binomial formulas, suitable for Grades 10-12.

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