To solve this binomial probability problem, let's break it down into the key elements needed for the binomial probability formula or using technology to calculate the probability:

Given Information:

• The sample size (number of trials): $n = 14$
• The number of successes (people who would vote by internet): $x = 11$
• The probability of success (people who would vote by internet, given by the survey): $p = 38\% = 0.38$
• The probability of failure (people who would not vote by internet): $q = 1 - p = 1 - 0.38 = 0.62$

Binomial Probability Formula:

$P(x = 11) = \binom{14}{11} \cdot (0.38)^{11} \cdot (0.62)^{3}$

Where: $\binom{14}{11} = \frac{14!}{11!(14-11)!} = \frac{14!}{11!3!}$ is the binomial coefficient (number of ways to choose 11 successes from 14 trials).

However, since the problem asks to use technology for the calculation, we can use a calculator or statistical software to compute the probability.

Using a Calculator or Statistical Software (like Excel, R, or Python):

The binomial probability formula can be computed using functions like BINOM.DIST in Excel or binom.pmf in Python. Using technology, you would input the following:

• Number of trials ($n = 14$)
• Number of successes ($x = 11$)
• Probability of success ($p = 0.38$)

For example, in Excel, you would use:

=BINOM.DIST(11, 14, 0.38, FALSE)

When computed, the result is approximately: $P(x = 11) \approx 0.0043$

Thus, the probability that exactly 11 of the 14 voters would prefer to vote online is approximately 0.0043 or 0.43%.

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Would you like further details on this calculation or help with parts (b) and (c)?

5 Related Questions:

1. How does changing the sample size affect the probability in a binomial distribution?
2. What is the probability that fewer than 11 voters would prefer online voting?
3. How can you calculate cumulative binomial probabilities using technology?
4. How does the binomial distribution compare to the normal distribution for large $n$?
5. What assumptions must hold for the binomial distribution to be valid?

Tip: When calculating binomial probabilities, make sure to correctly identify the probability of success $p$ and the number of trials $n$ for accurate results.