Question

1. From a box containing 10 red, 30 white, 20 blue and 15 orange balls, 5 are drawn at random. Find the probability that:
a) All are white - 0.008256723
b) Be red, white or blue - 0.009169617
c) They are neither red nor blue - 0.070788

2. A company will launch three new shampoos. It is believed that the probability that the first one is successful is 0.45, that the second one has it is 0.55 and that the third one has it is 0.75. What is the probability that
a) The three succeed? - 0.185625
b) At least two succeed? -
c) None succeed?

3. You have three boxes. In the first there are 6 rabbits and 2 pigeons; in the second, 8 rabbits and 5 pigeons, and in the third 4 rabbits and 6 pigeons. What is the probability of
a) take out a rabbit?
b) that the rabbit comes from the third box?
4. An exam has 10 multiple-choice questions. There are four possible answers for each question. Luis has not studied for the exam and has decided to do it at random. Assuming that you need 70% of the correct questions to pass, calculate the probability that Luis will pass the exam.

5. On average, four imperfections are observed in each 50m stretch of network cable. Calculate the probability that, in a 25 m section of this type of network cable:
a) More than three imperfections are observed b) At least one imperfection

6. According to one study, 30% of adults suffer from insomnia. If you find a group of randomly selected adults, what is the probability that:
a) the sixth revised is the first to suffer insomnia?
b) the fifth review is the third to suffer insomnia?

7. The weight of a newborn baby in a certain country is a random variable that follows a normal distribution with an average of 3.2 kg. And standard deviation of 0.4 kg.
a) Determine the percentage of newborn babies weighing 3.5 kg or more - 0.2266
b) Calculate the conditional probability that a newborn baby weighs more than 3.5 kg, if it is known to weigh at least 3 kg - 0.3277
c) From what weight is 10% of babies born weighing more? - 3.7126 Kg
   
8. In a hospital, 22% of transplants are cornea. Use the approximation to the binomial distribution by normal to calculate the probability that, in the following 120 transplants:
a) 30 are cornea - 0.0641
b) at least 30 are from the cornea - 0.1831

9. In a company of 5000 workers, it is desirable to know if the positive assessment of management management has varied greatly, which last year was conclusively concluded that it was 80% of the workers. For this, a sample of size 200 is carried out, resulting in the positive assessment being considered by 55% of the workers surveyed. Can we affirm that the valuation has varied with the probability of being 1% wrong? - Yes it has varied

10. A company producing pasteurized milk has as a rule not to accept raw milk with a fat content exceeding 34 g / 100g. A sample of 36 liters of milk obtained from as many cows belonging to the same farm gave an average value of fat content in milk of 35.2 g / 100g with a deviation of 4.1 g / 100g. Can milk be accepted by the pasteurizer? The company admits an error level of 1%. - Yes milk is accepted.
11. Calculate Zα / 2 in the following sections

a) With a confidence level of 96% - 2.05
b) With a confidence level of 99.5% and 92% - 2.81 and 1.75
c) with a level of significance of 2% and 7% - 2.32 and 1.81
d) With α = 0.03 and α = 0.11 - 2.17 and 1.6

12. A factory produces steel cables, whose resilience follows a normal distribution of unknown mean and standard deviation σ = 10 KJ / m3. A sample of 100 pieces was taken and through a statistical study a confidence interval (898.04 - 901.96) was obtained for the average resilience of the steel cables produced in the factory.

a) Calculate the value of the average resilience of the 100 pieces of the sample. - 900
b) Calculate the level of confidence with which this interval has been obtained. - 95% confidence
c) If you would like to have an error of 3 KJ / m3 in the confidence interval, what size should the sample be? A sample of

13. It is known that the average electric power consumption in a certain province is 721 Kwh.

A technology company in the region believes that its employees consume more than the provincial average. Collect information on the consumption of 15 randomly chosen employees, and obtain the following data:

710 774 814 768 823
732 675 755 770 660
654 757 736 677 797

If the distribution of monthly electricity consumption is normal:

a) Is there evidence to state that the average household electric power consumption of the company's employees is higher than the average consumption at the provincial level? Use a significance level of 10%. - Yes there is evidence.
b) What is the p-value of the decision? - 0.0878

14. An electric company manufactures cell phone batteries that have a duration that is distributed approximately normally with an average of 800 hours and a standard deviation of 40 hours. A random sample of 30 batteries has an average duration of 785 hours.

a) Do the data show sufficient evidence to say that the average duration is less than 800? Use a significance level of 5%. - Yes there is evidence to say that it is minor.
b) What is the probability of deciding that the average is 800 hours when in reality it is 780 hours? - 0.1357 (they don't come like this on the exam, it's difficult and they have to deduce several things)

15. A cake factory manufactures, in its usual production, 3% of defective cakes. A customer receives an order of 500 cakes from the factory. Calculate the probability that ...
a) Find more than 5% of defective cakes. - 0.004375
b) Find between 1% and 3% of defective cakes. - 0.4956
c) Calculate a 95% confidence interval for the percentage of defective cakes. (0.015-0.4495)

16. It is assumed that the distribution of the temperature of the human body in the population has an average of 37ºC and a standard deviation of 0.85ºC. A sample of 105 people is chosen, calculate the following probabilities:
a) That the average is less than or equal to 36.9 ºC - 0.114
b) That the average is greater than 38.5 ºC - 0
c) Find from what temperature 10% of the hottest bodies are found. - 38.09
d) Find from what temperature 5% of the coldest bodies are found. - 35.6

Answer 1

We would be looking at Question 1 all parts here.

We are given here that there are 10 red, 30 white, 20 blue and 15 orange balls in the bag. We are drawing 5 balls from these

a) Probability that all are white balls is computed here as:

= Number of ways to select 5 balls from 30 balls / Number of ways to select 5 balls from 75 balls

Therefore 0.0083 is the required probability here.

b)Probability that the balls are red, white or blue is computed here as:

= Number of ways to select 5 balls from (10 + 30 + 20) balls / Number of ways to select 5 balls from 75 balls

Therefore 0.3164 is the required probability here.

c) Probability that the balls are neither red nor blue

= Number of ways to select 5 balls from (30 + 15) balls / Number of ways to select 5 balls from 75 balls

Therefore 0.0708 is the required probability here.