To solve this problem, we need to use the binomial probability formula since we're dealing with a situation where there are only two possible outcomes (internet voting or not) for each selected voter.

The formula for binomial probability is:

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

Where:

• $n = 12$ (the number of likely voters selected)
• $k = 9$ (the number of voters who would vote via the internet)
• $p = 0.39$ (the probability of a voter choosing internet voting)
• $1 - p = 0.61$ (the probability of a voter not choosing internet voting)

Step 1: Calculate the binomial coefficient $\binom{n}{k}$

$\binom{12}{9} = \frac{12!}{9!(12 - 9)!} = \frac{12!}{9!3!}$

Step 2: Apply the binomial probability formula

Now, let's plug everything into the binomial formula:

$P(X = 9) = \binom{12}{9} (0.39)^9 (0.61)^3$

Let's calculate this value.

I'll compute it now. math

Given values

n = 12 # number of trials (voters) k = 9 # number of successes (internet voting) p = 0.39 # probability of success (internet voting)

Binomial coefficient (n choose k)

binomial_coeff = math.comb(n, k)

Calculate the binomial probability

probability = binomial_coeff * (pk) * ((1 - p)(n - k))

Round the result to five decimal places

round(probability, 5)The probability that exactly 9 of the 12 selected likely voters would choose to vote by internet is 0.01042 (rounded to five decimal places).

Would you like further explanation or assistance with any part of this calculation?

Here are five related questions that can deepen your understanding:

1. How would the probability change if 10 voters were selected instead of 12?
2. What happens to the probability when the percentage of internet voters changes to 50% (p = 0.50)?
3. How do you calculate the probability for a different number of successes (e.g., 7 or 10 voters choosing internet voting)?
4. How does the binomial distribution compare to the normal distribution in this case?
5. What is the cumulative probability of at most 9 voters choosing internet voting?

Tip: When calculating binomial probabilities, using the cumulative distribution function (CDF) or statistical software can simplify calculations for larger datasets or when dealing with complex binomial probabilities.