Let's go through each part and clarify the steps in solving this.

Part 1: Finding $P(1111)$

We need to calculate the binomial probability that exactly 1111 voters out of 1414 will vote by internet. The formula is:

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

Given:

• $n = 1414$ (number of trials),
• $x = 1111$ (number of successes),
• $p = 0.38$ (probability of success).

Using technology or software, we already found:

$P(1111) = 0.00207$

This is the probability that exactly 1111 voters will choose internet voting.

Part 2: Determining if 1111 voters is significantly high

To determine whether 1111 voters willing to vote by internet is a significantly high number, we need to calculate the probability of 1111 or more successes:

$P(x \geq 1111) = P(1111 \text{ or more})$

We already calculated:

$P(1111 \text{ or more}) = 0.00242$

Since $0.00242 \leq 0.05$, the number of 1111 voters is indeed considered significantly high because the probability of this happening is very small (less than 5%).

Part 3: Finding the probability of at least 1 voter choosing internet voting

Now, we need to calculate the probability that at least one of the selected likely voters would choose internet voting. The probability of at least one success can be found by subtracting the probability of zero successes from 1:

$P(\text{at least 1}) = 1 - P(0)$

Where $P(0)$ is the probability that no voters choose internet voting.

Using technology or software, you can calculate:

$P(0) = (1 - 0.38)^{1414} = (0.62)^{1414}$

Once computed, you get the result for $P(0)$, and then:

$P(\text{at least 1}) = 1 - P(0)$

After calculation, you should round your answer to three decimal places.

Would you like more details or have any further questions about any of the steps? Here are five related questions for further practice:

1. How do you calculate the expected value of a binomial distribution?
2. What does it mean for an event to be "significantly low" in terms of probability?
3. How would the results change if the probability of success, $p$, was higher?
4. What is the variance of the binomial distribution in this scenario?
5. Can you explain how to interpret cumulative binomial probabilities?

Tip: When dealing with binomial distributions, technology tools (like a calculator or software) are incredibly helpful for complex probabilities, but understanding the underlying formulas ensures deeper comprehension.