The binomial distribution

A new messaging app analysed that $30\,\%$ of sent messages contain at least one emoji. If a student sends a burst of $n = 6$ independent messages to a group, identify the parameters of the binomial distribution followed by the random variable $X$: "number of messages containing emojis" and write down the expression for its probability function $P(X = k)$.

Give reasons to state which of the following random variables follow a binomial distribution:

1. Drawing $4$ cards from a Spanish deck, one by one, without replacement, and counting the number of aces.

2. Rolling a fair die $10$ times and counting how many times an even number is obtained.

3. Asking $20$ people in a city for their favourite colour from a list of $5$ colours.

In a first-person shooter video game, a player has a probability of $80\,\%$ of hitting the target each time they shoot. If they fire a series of $n = 4$ independent shots, calculate the exact probability of hitting the target on exactly $3$ occasions using the binomial distribution formula.

Given a discrete random variable that follows a $X \sim B(n=5, p=0,4)$ distribution, complete the following table of its probability mass function for the indicated values:

On a photography social network, it is estimated that the probability of a follower "liking" a post by a given influencer is $p = 0,25$. If the influencer posts a story and $8$ followers view it independently, calculate the probability that it receives at least one "like".

Hint: Use the complementary event.

A wireless headphone manufacturer finds that $2\,\%$ of its devices leave the factory with a pairing defect. If they are packed in boxes of $10$ units, what is the probability that a box contains no defective headphones? Write only the formulation using combinations.

It is known that $15\,\%$ of subscribers to a streaming channel cancel their subscription before the end of the first month. If $100$ new subscribers are selected at random, calculate the expected number of cancellations (mean) and the standard deviation of this distribution.

A drone component manufacturing company carries out strict quality control. It has been observed that the number of defective sensors in batches of 20 units follows a binomial distribution. If the standard deviation of this distribution is exactly $1,2$ sensors:

1. a) Calculate the possible values of the probability $p$ that a sensor is defective.

1. b) If it is known that the probability of a sensor being non-defective is greater than that of it being defective, calculate the probability that in a batch of 20 units there are exactly 2 defective sensors.

1. c) For the value of $p$ found in the previous section, if a technician checks the sensors one by one, what is the probability that the first defective sensor appears precisely on the fourth check?

In a popular strategy video game, the probability that a "loot chest" contains a legendary skin is $15\%$. A player decides to open a pack of $8$ chests one after another. Assuming that the contents of each chest are independent of the others:

1. a) Identify the parameters of the binomial distribution followed by the random variable $X$: "number of legendary skins obtained".

1. b) Calculate the probability that the player obtains exactly $2$ legendary skins.

1. c) What is the probability that the player obtains no legendary skin in the whole pack?

A college league basketball player has a free-throw success rate of $80\%$. During a crucial game, she is awarded $10$ free throws. Assume the throws are independent and that the probability of success is the same on each one.

1. a) Define the random variable and its distribution.

1. b) Calculate the probability that she makes exactly $8$ of the $10$ throws.

1. c) If she needs to make at least $9$ of the $10$ shots to win the game, what is the probability that her team wins thanks to her free throws?

A cloud data storage server uses an integrity verification system for each packet of information sent. It is known that the probability that a packet is transmitted successfully without errors is $p$. If 10 packets are sent independently, the probability that at least one of them arrives with errors is $0,65132$.

1. a) Determine the value of the probability $p$ of successful transmission of a packet.

1. b) If the security of the channel is increased and the error probability per packet is reduced to one quarter of the original, what is the minimum number of packets that must be sent so that the probability that all of them arrive without errors is less than $0,5$?

The figure shows, as a guide, how the probability that all packets arrive without errors decreases with the improved channel as the number of packets $n$ increases, and compares it with the threshold $0,5$. The exact value of $n$ must be obtained by calculation.

On a microblogging social network, it is estimated that $15\%$ of the accounts are "bots". A researcher randomly selects a group of accounts for study.

1. a) What is the minimum size of the group of accounts that they must select to ensure, with a probability greater than $95\%$, that they will find at least one account that is a bot?

1. b) If the researcher finally decides to select a group of 12 accounts, what is the probability that the number of bots found lies within one standard deviation of the mean?

An electronic component manufacturing company knows that the probability of a capacitor being defective is $5\%$. The capacitors are packed in boxes for distribution.

1. a) If a box contains $20$ capacitors, what is the probability of finding exactly one defective component?
2. b) What is the probability that in a box of $20$ units there are at least two defective ones?
3. c) What is the minimum number of capacitors a box should contain so that the probability of finding at least one defective one is greater than $0,9$?

A pharmaceutical company claims that a new seasonal allergy treatment has an efficacy of $75\%$. The treatment is administered to a group of $9$ volunteer patients.

1. a) Calculate the probability that the treatment is effective in exactly $6$ patients.
2. b) What is the probability that the treatment fails in more than $2$ patients?
3. c) If the treatment is considered a commercial success if it works in at least $7$ of the $9$ patients, what is the probability of achieving such success?

A delivery app guarantees delivery in less than 30 minutes. The probability that a delivery driver meets this guarantee is $p=0,9$. A delivery driver makes 20 deliveries in a day.

1. a) Calculate the probability that the number of on-time deliveries is greater than 17, given that at least 16 were delivered on time.

1. b) If the cost to the company for each late delivery is $5$ euros and the profit for each on-time delivery is $2$ euros, calculate the expected value of the total profit after the 20 deliveries.

1. c) Determine the probability that the total profit is positive.