To solve this problem, we can use the binomial probability formula. The binomial distribution is appropriate because we have a fixed number of trials (14 likely voters), and each voter has only two possible outcomes: either they would vote via internet (success) or they would not (failure).

The binomial probability formula is:

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

where:

• $n$ is the number of trials (in this case, 14 voters),

• $k$ is the number of successes (in this case, 11 voters),

• $p$ is the probability of success (31% or 0.31),

• $\binom{n}{k}$ is the number of combinations of $n$ trials taken $k$ at a time, which is computed as:

  $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Step-by-step solution:

1. Given values: $n = 14, \quad k = 11, \quad p = 0.3131$

2. Calculate the binomial coefficient $\binom{n}{k}$: $\binom{14}{11} = \frac{14!}{11!(14-11)!} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364$

3. Calculate the probability: $P(X = 11) = 364 \cdot (0.3131)^{11} \cdot (1 - 0.3131)^{3}$

   First, calculate the powers: $(0.3131)^{11} \approx 5.1825 \times 10^{-6}, \quad (0.6869)^3 \approx 0.3240$

   Then, multiply these values: $P(X = 11) = 364 \cdot 5.1825 \times 10^{-6} \cdot 0.3240$ $P(X = 11) \approx 364 \times 1.6792 \times 10^{-6} = 6.111 \times 10^{-4}$

Thus, the probability that exactly 11 out of 14 likely voters would choose internet voting is approximately:

$P(X = 11) \approx 0.00061$

Final answer:

$P(X = 11) \approx 0.00061$

Would you like further details or have any questions? Here are five related questions to expand on this topic:

1. What is the probability that fewer than 11 voters would choose internet voting?
2. What is the expected number of voters who would choose internet voting?
3. How does the variance of this binomial distribution help in understanding voting patterns?
4. What is the probability that more than 11 voters would choose internet voting?
5. How does changing the sample size from 14 to 20 affect the probability?

Tip: When using binomial probability, double-check the calculation of powers and factorials for accuracy.