To solve this, we can use the binomial probability formula because the problem involves a fixed number of trials (8 consumers), each with two possible outcomes (either they believe cash will be obsolete or they don't), and the probability of success (believing that cash will be obsolete) is constant for each trial.

The binomial probability formula is:

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

Where:

• $n = 8$ (the number of trials, i.e., consumers)
• $k$ is the number of successes (the number of consumers who believe cash will be obsolete)
• $p = 0.396$ (the probability that a randomly selected consumer believes cash will be obsolete)
• $1 - p = 0.604$ (the probability that a randomly selected consumer does not believe cash will be obsolete)

You are asked to find the probability that fewer than 3 consumers believe cash will be obsolete, which means you need to find $P(X < 3)$. This can be broken down into the probabilities for $X = 0$, $X = 1$, and $X = 2$, and then summed:

$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$

Let's calculate each of these:

1. $P(X = 0)$: $P(X = 0) = \binom{8}{0} (0.396)^0 (0.604)^8$

2. $P(X = 1)$: $P(X = 1) = \binom{8}{1} (0.396)^1 (0.604)^7$

3. $P(X = 2)$: $P(X = 2) = \binom{8}{2} (0.396)^2 (0.604)^6$

Let me calculate these probabilities and find the total. math

Given values

n = 8 p = 0.396 q = 1 - p

Binomial probability formula

def binomial_prob(n, k, p, q): return math.comb(n, k) * (p ** k) * (q ** (n - k))

Calculate probabilities for X = 0, X = 1, X = 2

P_X_0 = binomial_prob(n, 0, p, q) P_X_1 = binomial_prob(n, 1, p, q) P_X_2 = binomial_prob(n, 2, p, q)

Total probability for fewer than 3

P_less_than_3 = P_X_0 + P_X_1 + P_X_2 round(P_less_than_3, 3)The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is approximately 0.324.

Would you like a more detailed explanation or have any further questions on this topic?

Here are some related questions you might find interesting:

1. What is the expected number of consumers who believe cash will be obsolete in the next 20 years?
2. How would the probability change if 12 consumers were selected instead of 8?
3. What is the variance of the number of consumers who believe cash will be obsolete?
4. How does the probability distribution of this scenario relate to a normal distribution as $n$ increases?
5. How do you calculate the probability for more than 3 consumers in this case?

Tip: The binomial distribution is a good model for situations where you're dealing with repeated trials, each with two possible outcomes, like "yes" or "no" questions.