39.7​% of consumers believe that cash will be obsolete in the next 20 years. Assume that 6 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 problem can be solved using the **binomial probability distribution** because there are only two possible outcomes for each consumer: either they believe that cash will be obsolete or they do not.

### Step-by-step Solution:

1. **Define Variables:**
   - \( n = 6 \): The number of trials (consumers selected).
   - \( p = 0.397 \): The probability that a consumer believes cash will be obsolete.
   - \( q = 1 - p = 0.603 \): The probability that a consumer does **not** believe cash will be obsolete.
   - We want to find the probability that **fewer than 3** consumers (i.e., 0, 1, or 2 consumers) believe cash will be obsolete.

2. **Binomial Probability Formula:**
   The binomial probability formula is:
   \[
   P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k}
   \]
   where:
   - \( \binom{n}{k} \) is the binomial coefficient: \( \frac{n!}{k!(n-k)!} \)
   - \( p^k \) is the probability of \( k \) successes (believers of cash obsolescence).
   - \( q^{n-k} \) is the probability of \( n-k \) failures (non-believers).

3. **Find the Probability for \( X < 3 \):**
   We need to calculate \( P(X = 0) + P(X = 1) + P(X = 2) \).

   **(i) \( P(X = 0) \):**
   \[
   P(X = 0) = \binom{6}{0} \cdot (0.397)^0 \cdot (0.603)^6 = 1 \cdot 1 \cdot (0.603)^6 \approx 0.0912
   \]

   **(ii) \( P(X = 1) \):**
   \[
   P(X = 1) = \binom{6}{1} \cdot (0.397)^1 \cdot (0.603)^5 = 6 \cdot 0.397 \cdot (0.603)^5 \approx 0.3590
   \]

   **(iii) \( P(X = 2) \):**
   \[
   P(X = 2) = \binom{6}{2} \cdot (0.397)^2 \cdot (0.603)^4 = 15 \cdot (0.397)^2 \cdot (0.603)^4 \approx 0.3968
   \]

4. **Sum the Probabilities:**
   \[
   P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.0912 + 0.3590 + 0.3968 = 0.847
   \]

### Final Answer:
The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately **0.847** or **84.7%**.

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Would you like further clarification on this solution or a breakdown of any step?

Here are 5 related questions for practice:
1. What is the probability that exactly 3 consumers believe that cash will be obsolete?
2. What is the probability that all 6 consumers believe cash will be obsolete?
3. How does the probability change if you select 10 consumers instead of 6?
4. What is the expected number of consumers who believe cash will be obsolete out of the 6 selected?
5. What is the variance and standard deviation for the number of consumers who believe cash will be obsolete?

**Tip:** For binomial problems, always check if the scenario involves a fixed number of independent trials with only two possible outcomes. This helps in identifying when to use the binomial distribution.

39.7% of consumers believe that cash will be obsolete in the next 20 years. Assume that 6 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.

Calculate the probability that fewer than 3 out of 6 randomly selected consumers believe that cash will be obsolete in the next 20 years. Learn how to use binomial distribution with step-by-step explanations and relevant formulas. Suitable for Grades 10-12.

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