​39.9​% 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.

This is a binomial probability problem where:

- The probability \( p \) of success (a consumer believing that cash will be obsolete) is \( p = 0.399 \).
- The number of trials \( n \) is \( 7 \).
- We are interested in finding the probability that fewer than 3 consumers (i.e., 0, 1, or 2) believe that cash will be obsolete.

The binomial probability formula is:

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

Where:
- \( P(X = k) \) is the probability of exactly \( k \) successes in \( n \) trials.
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.
- \( p \) is the probability of a success.
- \( 1 - p \) is the probability of a failure.

To find the probability that fewer than 3 consumers believe that cash will be obsolete, we calculate \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \).

### Step-by-step Calculation:

1. **Calculate \( P(X = 0) \):**
   \[
   P(X = 0) = \binom{7}{0} (0.399)^0 (0.601)^7 = 1 \times 1 \times (0.601)^7 \approx 0.042
   \]

2. **Calculate \( P(X = 1) \):**
   \[
   P(X = 1) = \binom{7}{1} (0.399)^1 (0.601)^6 = 7 \times 0.399 \times (0.601)^6 \approx 0.196
   \]

3. **Calculate \( P(X = 2) \):**
   \[
   P(X = 2) = \binom{7}{2} (0.399)^2 (0.601)^5 = 21 \times (0.399)^2 \times (0.601)^5 \approx 0.334
   \]

### Adding These Probabilities:
\[
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.042 + 0.196 + 0.334 = 0.572
\]

So, the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately **0.572**.

Would you like a deeper explanation of any part of this calculation?

### Related Questions:
1. What is the probability that exactly 4 consumers believe that cash will be obsolete?
2. How would the probability change if the sample size were 10 instead of 7?
3. What is the probability that more than 5 consumers believe cash will be obsolete in the next 20 years?
4. How would the probability change if the percentage of consumers believing in cash obsolescence increased to 50%?
5. Can you find the mean and standard deviation of the distribution?

### Tip:
When dealing with binomial distributions, remember that probabilities add up, so breaking down problems into smaller events (e.g., fewer than or greater than a number) can help simplify calculations.

39.9% 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.

Discover how to calculate the probability that fewer than 3 out of 7 consumers believe that cash will be obsolete in 20 years using the binomial probability formula. Learn key probability concepts, binomial distribution, and step-by-step calculations. Suitable for Grades 10-12.

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