Solve The Following Exercises

Solve The Following Exercises

Instructions: Solve the following exercises: 1. A study of queues to pay in the cash registers at a supermarket showed that during the busiest hour the average number of customer waiting to pay is 4. What is the probability that during that period: a. There are no customers waiting? b. Four customers are waiting in the line? c. Four or less customers are waiting in the line? d. Four or more customers are waiting? 2. Classes of first year in a school are full and it is necessary to open new groups. We consider that half of the new students will be boys and the second half will be girls. The size of each group is limited to 18 students. a. What is the probability that exactly 9 boys and 9 girls will be in a group? b. What is the probability that 11 or more boys will be in the new group? Is this the same probability as there will be less than 7 girls in the group? c. What is the probability that exactly 7 girls and 11 boys will be in the new group? d. What is the probability that on the first day of classes when the doors are opened the first student to arrive to the classroom will be a girl? Solve the following exercises: 1. A food company offers 3 different meals: sopa, caldo and rice. The prices of the meals are $15.00, $25.00 y $22.00 respectively, based on an study the customers ask for the meals in this way, 27% soup, 55% caldo and 18% rice. a. Is this a discrete probability distribution? Why? b. Calculate the average amount charged per meal c. What is the variance in the charge of the meals? d. What is the standard deviation? Interprete the results. e. Graph the probability distribution Ratio of change Number of days - 1 / / / / / . A trader observed the behaviour of certain shares during 50 days in the Mexican stock market and he registered the following data: Ratio of change Number of days - 1 / / / / / . a. Does this information correspond to a discrete probability distribution? b. Transform this information to a probability distribution and calculate the mean and variance. Interprete the results. c. Create the graph that represent this probability distribution. 4. The manager of Human Resources in a company has information about the accidents at work during one month. He created the following probability distribution: Number of accidents Probability .......05 a. Calculate the mean, variance and standard deviation b. Interprete the results c. Create the graph that represents the probability distribution. 5. A bakery offers cakes with special decoration and also standard cakes. The owner has created the following probability distribution: Sold cakes in one day Percentage % % % % % Calculate the mean, variance and standard deviation. Interprete the results Create the graph that represents the probability distribution. Instructions: Solve the following exercises: 1. In one experiment a person throws two dices four times. a. What is the probability that in the last throw the person gets a 12? b. What is the probability that in the last throw the person gets a number higher than 10? 2. The owner of a flower shop has 18 delivery vehicles. Suppose that 4 of the 18 vehicles have air conditioning to preserve the flowers. Four vehicles are chosen to test them. What is the probability that all the chosen vehicles have air conditioning to preserve the flowers? 3. An electronic devices shop received a shipment of 100 DVDs. The owner reported that there were 30 defective devices. They decided to test 2 DVDs. What is the probability that both are defective?

Paper For Above instruction

This comprehensive analysis investigates key probabilistic and statistical scenarios encountered in various real-world contexts, from queue management at supermarkets to quality control in manufacturing and investment behaviors in financial markets. The primary objective is to articulate the underlying principles of probability distribution models, their calculations, and interpretations, emphasizing their practical applications and implications.

Queue Management at Supermarkets:

The scenario involves understanding customer wait times represented through the Poisson distribution, suitable for modeling count data such as the number of customers waiting in line. Given an average rate (λ) of 4 customers during the busiest hour, Poisson probabilities facilitate the calculation of the likelihood of different queue lengths:

- The probability of no customers waiting (k=0): P(0) = e^(-λ) = e^(-4) ≈ 0.0183, indicating a low chance of an empty queue in peak hours.

- Exactly four customers waiting (k=4): P(4) = (λ^4 * e^(-λ))/4! ≈ 0.0916, sensitive to fluctuations in customer load.

- Four or fewer customers (k ≤ 4): Sum of probabilities from 0 to 4, approximately 0.430, informing resource allocation.

- Four or more customers (k ≥ 4): Complement of P(k≤3), approximately 0.57, highlighting potential congestion risks.

This modeling aids store managers in staffing and optimizing customer service strategies effectively.

Classroom Group Composition:

Applying the binomial distribution captures the probability of student gender compositions per group. For a group of 18 students, assuming each student has a 0.5 chance of being a boy or girl:

- The probability of exactly 9 boys and 9 girls (k=9): P = C(18, 9) * (0.5)^18 ≈ 0.193.

- The probability of 11 or more boys: sum of P(k) for k=11..18, which involves calculating binomial probabilities; notably, P(k ≥ 11) ≈ 0.35.

Interestingly, the probability of having fewer than 7 girls is identical, reflecting symmetry around k=9.

- Probability of exactly 7 girls (and 11 boys): Similar to the first, approximately 0.163.

- For the first student arriving as a girl: probability is 0.5, assuming random and independent arrivals.

These insights assist educators in planning class sizes and understanding demographic distributions.

Menu Preference Analysis:

The discrete probability distribution of meal requests at a food company reflects customer preferences:

- Probabilities: Soup (27%), Caldo (55%), Rice (18%)

- Confirmed as a valid distribution because probabilities sum to 1.

- The expected income per meal:

E = (0.27 15) + (0.55 25) + (0.18 * 22) ≈ 20.46 dollars.

- Variance calculations involve the squared deviations, resulting in a variance of approximately 24.35, with the standard deviation about 4.93 dollars.

These metrics provide operational insights such as revenue variability and forecasting.

Stock Market Behavior:

Analyzing share change ratios over 50 days entails transforming raw data into a probability distribution:

- Assuming a frequency distribution of daily ratios, the data can be modeled using discrete probabilities.

- Calculations of the mean and variance reveal the expected daily change and dispersion, aiding investors in risk assessment.

- Graphical representations effectively communicate the distribution’s shape, skewness, and dispersion.

This analysis supports strategic decision-making in investments.

Workplace Safety:

A probability distribution of work-related accidents demonstrates the variability in safety performance:

- With a probability of 0.05 for certain accident counts, calculations of mean and variance disclose safety trends.

- The mean number of accidents helps gauge safety levels, while the variance indicates stabilization or variability.

- Graphical depiction offers visual insight into accident distribution and risk profiles.

This statistical profile informs safety protocols and resource allocation for accident prevention.

Bakery Sales Probability:

Modeling the daily sales of decorated versus standard cakes through probability distribution:

- The percentages indicate sales likelihoods, enabling calculation of mean, variance, and standard deviation.

- These metrics facilitate inventory planning and sales forecasting.

- Visualizing the distribution supports marketing strategies and operational adjustments.

Dice Experiments:

The probability of specific outcomes in dice throws exemplifies classic probability applications:

- Probability of rolling a sum of 12 (double sixes): 1/36, derived from the fact that only one combination yields a 12.

- Probability of rolling a number higher than 10: includes 11 and 12, with combined probability of 3/36 or 1/12.

These fundamental probabilities underpin many probabilistic models.

Vehicle Testing and Quality Control:

Calculating probabilities in selection processes:

- The chance all tested vehicles have air conditioning involves hypergeometric distribution:

P = (4/18)(3/17)(2/16)*(1/15) ≈ 0.00063.

- The probability both DVDs are defective from a shipment of 100 with 30 defective units:

P = (30/100) * (29/99) ≈ 0.0877.

These computed probabilities assist in operational decision-making and quality assurance.

Overall, mastering probability distributions and statistical measures enables effective decision-making in diverse industries and scenarios, supporting operational efficiency, risk management, and strategic planning.

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