Let's walk through this problem step by step, starting with part (a).

Part 1: a. Calculating the Probability that Exactly 10 Out of 13 Voters Would Choose Internet Voting

The formula for binomial probability is given by:

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

Where:

• $n = 13$ (the number of voters)
• $x = 10$ (the number of voters choosing internet voting)
• $p = 0.34$ (the probability of one voter choosing internet voting)
• $q = 1 - p = 0.66$ (the probability of one voter not choosing internet voting)

Now, using technology (such as a binomial probability calculator or software), the probability of exactly 10 voters choosing internet voting is calculated as follows:

$P(10) = \binom{13}{10} (0.34)^{10} (0.66)^{3}$

The result is approximately:

$P(10) \approx 0.00170$

So, the probability is:

$P(10) = 0.00170$

Part 6: b. Is 10 Significantly High?

We are given that a number of successes is considered significantly high if the probability of obtaining that number of successes or more is less than or equal to 0.05.

Now we need to calculate the probability of getting 10 or more voters choosing internet voting, which is:

$P(10 \text{ or more}) = P(10) + P(11) + P(12) + P(13)$

Using technology to calculate the individual probabilities for $P(11), P(12),$ and $P(13)$ and summing them up, we find that:

$P(10 \text{ or more}) \approx 0.00229$

Since $P(10 \text{ or more}) = 0.00229$ is less than 0.05, having 10 or more voters choosing internet voting is considered significantly high.

Summary:

• $P(10) = 0.00170$
• $P(10 \text{ or more}) = 0.00229$, which is less than 0.05, so 10 is considered a significantly high number of successes.

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Do you want more details or have any questions on this explanation?

Here are 5 related questions to extend your understanding:

1. How would the probability change if only 5 voters out of 13 choose internet voting?
2. What happens to the probability if we change $p$ to 0.50 instead of 0.34?
3. How does the probability of 10 or more successes change if we increase the number of trials $n$?
4. Can you explain how to calculate $\binom{n}{x}$ in a binomial probability formula?
5. What does it mean if a number of successes is considered significantly low?

Tip: For binomial distributions, the larger the number of trials $n$, the narrower the probabilities become around the expected number $np$.