The Admission Cost For A Movie Premiere At The Local Theater

The Admission Cost For A Movie Premier At The Local Theater Was Is 8

The problem involves two main questions: determining the number of children and adults attending a movie premiere based on total attendance and revenue, and calculating the probability that exactly two out of four randomly selected workers are union members, given the union membership rate.

First, consider the scenario of the movie premiere. The variables are defined as follows:

- x = number of adults

- y = number of children

According to the problem, the total number of people attending the premiere was 250, which can be expressed as:

\[ x + y = 250 \]

The total revenue generated was \$2,150. Given the ticket prices ($10 for adults and $8 for children), the revenue equation becomes:

\[ 10x + 8y = 2150 \]

To find the values of x and y, we'll use substitution or elimination methods. Starting with the first equation, express y as:

\[ y = 250 - x \]

Substitute this into the revenue equation:

\[ 10x + 8(250 - x) = 2150 \]

\[ 10x + 2000 - 8x = 2150 \]

\[ (10x - 8x) + 2000 = 2150 \]

\[ 2x + 2000 = 2150 \]

\[ 2x = 150 \]

\[ x = 75 \]

Now, determine y:

\[ y = 250 - 75 = 175 \]

Therefore, 75 adults and 175 children attended the premiere.

Next, analyze the probability question regarding union membership. The probability that a randomly selected worker is a union member is 96%, or 0.96. When selecting 4 workers at random, the probability that exactly 2 are union members can be modeled using the binomial probability formula:

\[ P(k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:

- n = 4 (number of trials)

- k = 2 (number of successes, union members)

- p = 0.96 (probability of success)

Calculating:

\[ P(2) = \binom{4}{2} (0.96)^2 (0.04)^2 \]

The binomial coefficient:

\[ \binom{4}{2} = \frac{4!}{2! \times 2!} = 6 \]

Substituting:

\[ P(2) = 6 \times (0.96)^2 \times (0.04)^2 \]

\[ P(2) = 6 \times 0.9216 \times 0.0016 \]

\[ P(2) = 6 \times 0.00147456 \]

\[ P(2) \approx 0.00884736 \]

Expressed as a percentage, the probability is approximately 0.88%.

In conclusion, based on the problem data, 75 adults and 175 children attended the movie premiere, and the probability that exactly two out of four randomly selected workers are union members is approximately 0.88%.

Paper For Above instruction

The Admission Cost For A Movie Premier At The Local Theater Was Is 8

This paper provides a detailed analysis of two interconnected problems: the calculation of attendance figures at a local movie premiere based on ticket sales and total revenue, and the determination of the probability that exactly two workers out of four randomly chosen are union members, given the union membership rate at a university. These problems exemplify the application of algebraic methods and probability theory in real-world scenarios.

Analysis of Movie Premiere Attendance

The first problem involves establishing the number of children and adults attending the premiere. We are given the total attendance (250 people) and the total revenue generated (\$2,150). The ticket prices are specified: \$10 for adults and \$8 for children. To analyze this, we define the variables:

• x = number of adults
• y = number of children

Using these variables, the total attendance equation can be formulated as:

x + y = 250

Similarly, the total revenue equation becomes:

10x + 8y = 2150

Substituting y from the first equation into the revenue equation yields:

10x + 8(250 - x) = 2150

Expanding the terms gives:

10x + 2000 - 8x = 2150

Combining like terms leads to:

2x + 2000 = 2150

Subtracting 2000 from both sides:

2x = 150

Dividing both sides by 2 yields:

x = 75

Substituting back into the attendance equation to find y:

y = 250 - 75 = 175

Therefore, the analysis indicates that 75 adults and 175 children attended the movie premiere.

Probability of Selecting Union Members

The second problem deals with probability theory, specifically the likelihood of selecting exactly two union members from four randomly chosen workers. The university reports that 96% of its workers are union members, meaning:

p = 0.96

Using the binomial probability distribution, the probability of exactly k successes (union members) in n trials (workers selected) is given by:

P(k) = ⁿC<sub>k</sub&gt> p^k (1 - p)^{n - k}

Here, n = 4, k = 2. Calculating the binomial coefficient:

⁴C<sub>2</sub&gt> = 6

Computing the probability:

P(2) = 6 × (0.96)² × (0.04)²

Calculating the powers:

• (0.96)² ≈ 0.9216
• (0.04)² = 0.0016

Multiplying these gives:

6 × 0.9216 × 0.0016 ≈ 6 × 0.00147456 ≈ 0.00884736

This probability translates to approximately 0.88%, indicating that there's less than a 1% chance that exactly two out of four workers selected are union members, considering the high union membership rate.

Conclusion

The detailed calculations elucidate that a significant majority of the movie premiere's attendees were children, with 175 present, and a smaller subset of adults, totaling 75. This distribution aligns with typical family-oriented event demographics. In terms of union representation, the probability assessment reveals the rarity of selecting exactly two union members among four randomly chosen workers due to the high overall union membership rate (96%). These analyses demonstrate how algebra and probability can effectively interpret real-world data, aiding in strategic decision-making and resource planning.

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