The binomial distribution

In a survey on digital consumption habits, it was determined that the probability that a Baccalaureate student prefers using a tablet instead of a computer to study is $p = 0,3$. If we randomly select $8$ students from a class:

1. a) Identify the parameters $n$ and $p$ of the binomial distribution that models this situation.
2. b) What is the probability that exactly $3$ of them prefer the tablet?

A multiple-choice exam consists of $10$ questions with $4$ options each, of which only one is correct. A student who has not studied decides to answer all the questions at random.

1. a) Justify why this situation follows a binomial distribution $B(n, p)$.
2. b) What is the probability that they get exactly $5$ questions right?

A content creator on TikTok knows that the probability of one of their videos going "viral" (reaching more than 10 000 views) is $15\%$. If they have published $5$ independent videos this week:

1. a) Calculate the probability that none of the videos go viral.
2. b) Calculate the probability that exactly $2$ videos do.

A basketball player shooting free throws has a probability of scoring of $80\%$. If they shoot a series of $4$ independent throws:

1. a) Construct the table of the probability function for the possible successes ($0, 1, 2, 3$ and $4$ baskets).
2. b) What is the most likely outcome?

You can do it on paper and send a photo with your answer.

It is known that $60\%$ of young people in a city use public transport to go to school. If we choose a group of $6$ young people at random, calculate the probability that:

1. a) At least one of them uses public transport.
2. b) All of them use public transport.

In a nature reserve, it has been observed that $25\%$ of a migratory bird species have a white patch on their wing. If a photographer takes photos of a flock of $12$ birds:

1. a) What is the probability that none of them have the white patch?
2. b) What is the probability that exactly one has the patch?

In a quality control check at an electric scooter component factory, it has been estimated that $4\%$ of the parts have a minor cosmetic defect. If batches of $50$ parts are analysed:

1. a) What is the mean number of defective parts we expect to find per batch?
2. b) Calculate the standard deviation of the distribution.

A language app sends daily notifications to its users. The probability that a user opens the notification is $p = 0,4$. If a user receives $7$ notifications over the course of a week:

1. a) Calculate the probability that they open exactly $4$.
2. b) How many notifications are expected to be opened on average that week?

A drone component manufacturing company knows that the probability of a motor being defective is $p$. They are packed in boxes of $8$ units. The quality control department has observed that the probability of finding exactly $2$ defective motors in a box is three times the probability of finding exactly $1$ defective motor.

1. a) Determine the value of parameter $p$.
2. b) Calculate the expected value and standard deviation of the number of defective motors in an order of $100$ motors.
3. c) If a customer buys a box of $8$ motors, what is the probability that they must return it because it contains more than $2$ defective motors?

A cybersecurity analyst has determined that the probability of an unauthorised access attempt to a corporate server being detected by the firewall is $p$. It is known that if $10$ independent attempts are made, the probability that no attempt is detected is $0,0563$.

1. a) Calculate the value of the detection probability $p$.
2. b) If $20$ access attempts occur in a day, what is the probability that at least $18$ of them are detected?
3. c) What is the minimum number of attempts an attacker should make so that the probability of at least one of them not being detected is greater than $0,99$?

The final exam of a subject is a multiple-choice test and consists of $10$ questions with $4$ answer options each, of which only one is correct. A student answers all the questions at random. To pass, a grade of at least $5$ is required, knowing that each correct answer adds $1$ point and each incorrect answer subtracts $0,25$ points.

1. a) Define the random variable $X$ that counts the student's correct answers, state how it is distributed and what its parameters are.

1. b) Determine the minimum number of correct answers needed to pass the exam.

1. c) Calculate the probability that the student passes the exam by answering completely at random.

A television channel broadcasts a game show where contestants must choose between two boxes, one with a prize and the other empty. A contestant takes part in the show for $n$ consecutive days. The probability that they win the prize at least once in those $n$ days is $0,999$.

1. a) Calculate the minimum value of $n$.
2. b) For that value of $n$, calculate the probability that they win the prize on exactly half of the days.
3. c) If each prize has a value of $500$ €, calculate the contestant's expected winnings and the probability that their total winnings exceed $2000$ €.