Exercises: Binomial coefficients

To organize the senior class graduation party, an events committee made up of $3$ students must be chosen. If there are $20$ student volunteers in the class, in how many different ways can this group be chosen? Use the formula for binomial coefficients:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Calculate and simplify the value of the following expressions without using a calculator, applying the properties of factorials:

1. a) $\frac{8!}{6!}$

1. b) $\binom{7}{2}$

1. c) $5! - 4!$

Before studying the binomial distribution, it is essential to distinguish the type of variable. Classify the following random variables as discrete or continuous:

1. The number of "likes" an Instagram post receives in an hour.

2. The exact time it takes a student to get to school.

3. The number of people who test positive on an antigen test in a sample of $50$ individuals.

4. The height of the players on the US national basketball team.

A study states that $70\%$ of American teenagers use TikTok daily. If we randomly select a group of $8$ teenagers from a high school:

1. a) Identify the parameters $n$ and $p$ of the binomial distribution that this situation follows.
2. b) Define the random variable $X$ associated with this experiment.
3. c) What is the probability that exactly all $8$ teenagers use the app? (Express only the setup of the formula).

Knowing that for a binomial distribution $B(n, p)$, the probability of obtaining $k$ successes is:

$$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$$

Calculate the probability of obtaining exactly $2$ heads when flipping a fair coin $4$ times.

In the section on binomial coefficients, there is a symmetry property which states that $\binom{n}{r} = \binom{n}{n-r}$.

Verify this property by calculating the values of $\binom{10}{8}$ and $\binom{10}{2}$. What conclusion do you draw regarding the calculation effort when $r$ is a number close to $n$?

A multiple-choice exam consists of $5$ true or false questions. If a student answers at random (i.e., $p=0,5$):

1. a) What is the probability that they do not get any questions right ($P(X=0)$)?
2. b) Using the complementary event, set up how you would calculate the probability that they get at least one question right.

In a binomial distribution $X \sim B(100; 0,2)$, calculate:

1. a) The mean or expected value ($\mu = n \cdot p$).
2. b) The variance ($\sigma^2 = n \cdot p \cdot q$).
3. c) The standard deviation ($\sigma = \sqrt{n \cdot p \cdot q}$).

Briefly explain what the value of the mean signifies in this context.

In a high-performance sports center, they want to select a group of 4 athletes to represent the United States in an international relay competition. If there are 12 athletes available at the center (7 men and 5 women), calculate:

1. a) The total number of different ways in which the team of 4 people can be chosen.
2. b) In how many of those combinations would the team be formed by exactly 2 men and 2 women?
3. c) Use the properties of binomial coefficients to prove, without calculating the numerical value, that:

$$\binom{12}{4} = \binom{11}{3} + \binom{11}{4}$$

A young entrepreneur has launched a sustainable fashion app. According to her market research, $35\%$ of users who download the app end up making a purchase in the first week. If 10 new independent users have registered today:

1. a) Define the random variable $X$ and justify whether it follows a binomial distribution, indicating its parameters.
2. b) Calculate the probability that exactly 4 of these users make a purchase.
3. c) What is the probability that, at most, 2 users buy in the first week?

It is known that $15\%$ of the containers produced by a plastic recycling plant do not meet the required purity standards. A quality inspector randomly selects 8 containers for analysis.

1. a) Calculate the probability that all the selected containers are suitable (meet the standards).
2. b) Calculate the probability that at least one of the containers is defective.
3. c) What is the probability that there are more than 6 suitable containers in the sample?

A multiple-choice exam consists of 12 questions, each with 4 answer options of which only one is correct. A student who has not studied at all decides to answer all the questions at random.

1. a) Calculate the expected number of correct answers (mean) and the standard deviation of this distribution.
2. b) What is the probability that the student passes the exam (obtains at least 6 correct answers)?
3. c) If the exam had 100 questions, what parameters would the resulting binomial distribution have? (Do not perform the probability calculations).

In a vaccination campaign at a high school, it has been observed that $10\%$ of the students experience some mild side effect (such as a sore arm). If a class with 25 vaccinated students is selected:

1. a) What is the mean number of students expected to have side effects?
2. b) Calculate the probability that the number of students with side effects is less than the mean.
3. c) What is the probability that exactly 3 students have side effects?

Solve the following questions related to combinatorics and factorials:

1. a) Find the value of $n$ in the following equation:

$$\binom{n}{2} = 21$$

1. b) Simplify the following expression, leaving the result in terms of $n$:

$$\frac{(n+1)! + n!}{(n-1)!}$$

1. c) How many 5-digit numbers can be formed using only the digits $\{1, 2, 3, 4, 5\}$ if no digit is allowed to be repeated? How would the result change if we allow repetitions?

A content creator on TikTok knows that the probability of one of their videos going "viral" (exceeding 100,000 views) is 0.05. If they decide to post one video a day for a week (7 days):

1. a) What is the probability that none of their videos go viral?
2. b) Calculate the probability that at least 2 videos are viral.
3. c) Represent graphically using a tree diagram the possibilities of success (V) and failure (N) for the first 3 days and assign the probabilities to each branch.

A courier company uses drones to deliver parcels in rural areas. It has been proven that the probability of a drone having a precision error upon landing is 0.12. If a drone makes 15 landings in a day:

1. a) Calculate the probability that it is successful in all landings.
2. b) Calculate the probability that it has errors in more than 2 landings.
3. c) If the company wants the mean number of errors per day to be at most 1, how many landings should the drone make per day following the same error rate?

In an e-sports tournament, the probability of a professional "League of Legends" player achieving a "Pentakill" in a competitive match is $p = 0,008$. A team decides to participate in a league consisting of a certain number of matches $n$. It is known that the probability of the team not achieving any "Pentakills" in the entire league is less than $0,6$.

1. a) Determine, using logarithms, the minimum number of matches the team must play for this condition to be met.

1. b) If they finally play $120$ matches, calculate the probability that they achieve exactly two "Pentakills", expressing the final result to four decimal places. To calculate the binomial coefficient, use the factorial formula: $\displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}$.

A well-known social network is analyzing the behavior of its "stories". It has been proven that the distribution of the number of users who interact with an ad follows a binomial model. In a sample of users, it is known that the mean number of interactions is $\mu = 12$ and the variance is $\sigma^2 = 7,2$.

1. a) Calculate the sample size $n$ and the probability of interaction $p$ for each user.

1. b) Once $n$ and $p$ have been obtained, determine the probability that, in a new sample of the same size, the number of interactions is greater than $2$ but less than or equal to $4$.

A smartphone component manufacturing company carries out quality control. Each production batch of a critical part contains $2\%$ defective units. The parts are packed in boxes of $15$ units. A customer buys a shipment of $100$ boxes.

1. a) What is the probability that in a randomly chosen box there are at least $2$ defective parts?

1. b) If the customer returns the box if it contains more than one defective part, how many boxes are expected to be returned out of the $100$ they bought?

In the mathematics department of a high school, they want to form a graduation committee. There are $12$ Science students and $10$ Social Sciences students.

1. a) In how many ways can a committee of $6$ people be chosen if there must be at least $4$ Social Sciences students? Use the properties of binomial coefficients to simplify the manual calculations.

1. b) If the selection process for each member were independent and the probability of choosing a Social Sciences student were the actual proportion in the group ($10/22$), what would be the probability that in a committee of $6$ people chosen with replacement there would be exactly $3$ of each subject area?

A study on reading habits in 17-year-old teenagers states that $70\%$ read regularly in digital format. $10$ teenagers are selected at random.

1. a) Calculate the probability that the number of teenagers who read digitally is equal to the mode of this binomial distribution. (Note: The mode in a $B(n, p)$ is the integer value $k$ such that $(n+1)p - 1 \leq k \leq (n+1)p$).

1. b) If it is known that at least $8$ of them read in digital format, what is the probability that all $10$ do so? (Conditional probability).

Prove, using the definition of a binomial coefficient using factorials, the following identity (known as Pascal's rule or Stifel's property):

$$\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$$

Apply this result to calculate the value of $\displaystyle \binom{10}{4} + \binom{10}{3}$ without calculating each term separately.

A civil service exam consists of $80$ multiple-choice questions with $4$ options each, of which only one is correct. A candidate answers $12$ questions they did not know at random.

1. a) Let $X$ be the variable "number of correct answers in the $12$ questions". Define the parameters of the distribution.

1. b) Calculate the probability that they have more correct answers than incorrect ones.

1. c) If they need at least one correct answer to pass that section, calculate the probability that they fail.

A delivery app has a probability of delivery delay of $p = 0,15$. A delivery driver makes $20$ deliveries in a night shift.

1. a) What is the probability that they have exactly $3$ delays? Express the binomial coefficient as a product of fractions before calculating.

1. b) If the delivery driver receives a penalty if they have more than $4$ delays, what is the probability of being penalized? (Set up the calculation using the complementary event and justify why it is more efficient in this case).